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arxiv: 2604.09379 · v1 · submitted 2026-04-10 · ⚛️ physics.optics

Raman amplification and ISRS in SDM links: Analytical evaluation and closed-form models for optical transmission

Lucas Alves Zischler , Antonio Mecozzi , Cristian Antonelli This is my paper

Pith reviewed 2026-05-10 16:59 UTC · model grok-4.3

classification ⚛️ physics.optics
keywords Raman amplificationISRSSDMdistributed Raman amplificationmode couplinginter-modal effectsclosed-form modelsoptical fibers
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The pith

Closed-form expressions for Raman amplification and ISRS in SDM links are derived, including mode coupling and inter-modal effects.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes closed-form models for distributed Raman amplification (DRA) and inter-channel stimulated Raman scattering (ISRS) in space-division multiplexing (SDM) fibers. These models apply to designs with arbitrarily-coupled degenerate mode-groups and incorporate mode coupling along with inter-modal nonlinear effects. They extend single-mode fiber approaches to multi-mode SDM links and show strong agreement with numerical simulations. This matters for engineers designing high-capacity optical networks, as the expressions allow quick evaluation of gain and mode-dependent gain without heavy computation. The work also provides methods to estimate Raman response profiles experimentally in SDM fibers.

Core claim

In this work, we expand upon previous literature by providing closed-form expressions modelling DRA and ISRS in common SDM fiber designs that support arbitrarily-coupled degenerate mode-groups, incorporate mode coupling, and accounting for inter-modal non-linear effects, showing excellent agreement with simulations. The derived formulas are then applied to representative scenarios, illustrating how distinct pump and fiber configurations influence gain and mode-dependent gain (MDG). Finally, we describe suggested routines for experimentally estimating the Raman response profiles of SDM fibers.

What carries the argument

Closed-form analytical expressions for DRA and ISRS that account for mode coupling in arbitrarily-coupled degenerate mode groups of SDM fibers.

If this is right

  • The expressions enable prediction of gain and mode-dependent gain for various pump and fiber configurations in SDM links.
  • Inter-modal nonlinear effects are incorporated in the models for accurate SDM transmission analysis.
  • Suggested experimental routines allow estimation of Raman response profiles in SDM fibers.
  • The models facilitate evaluation of how different configurations affect Raman amplification in multi-mode systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These closed-form models could simplify the simulation and optimization of large SDM networks by reducing reliance on numerical methods.
  • The framework might be adapted to analyze other nonlinear optical effects in SDM fibers beyond Raman.
  • Mode-dependent gain control strategies could be developed using these expressions to improve SDM link performance.
  • Integration into optical system design tools would support practical deployment of SDM technology.

Load-bearing premise

The derivations assume that common SDM fiber designs can be represented by arbitrarily-coupled degenerate mode-groups and that inter-modal nonlinear effects admit closed-form treatment under the same approximations used for single-mode fibers.

What would settle it

Experimental measurements or detailed simulations on an SDM fiber with strong mode coupling that deviate substantially from the closed-form predicted gain and MDG would falsify the applicability of the models.

Figures

Figures reproduced from arXiv: 2604.09379 by Antonio Mecozzi, Cristian Antonelli, Lucas Alves Zischler.

Figure 1
Figure 1. Figure 1: Refractive index profiles, lateral field power distribu [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Mode- and mode-group-dependent parameters of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Longitudinal profiles of the mode-group-averaged signal powers, normalized to the fundamental mode, with a Raman [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Longitudinal profiles of mode-group-averaged powers, [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Longitudinal power profile of a signal launched in [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Longitudinal profiles of mode-group-averaged ASE [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 9
Figure 9. Figure 9: (a) Approximation error versus distance, where line [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
read the original abstract

In optical communications, the Raman effect is exploited for its lasing properties in distributed Raman amplification (DRA) and leads to spectral distortions through inter-channel stimulated Raman scattering (ISRS). In single-mode fibers, these effects are well understood and modeled, but equivalent closed-form expressions for arbitrarily coupled space-division multiplexing (SDM) links are lacking. In this work, we expand upon previous literature by providing closed-form expressions modelling DRA and ISRS in common SDM fiber designs that support arbitrarily-coupled degenerate mode-groups, incorporate mode coupling, and accounting for inter-modal non-linear effects, showing excellent agreement with simulations. The derived formulas are then applied to representative scenarios, illustrating how distinct pump and fiber configurations influence gain and mode-dependent gain (MDG). Finally, we describe suggested routines for experimentally estimating the Raman response profiles of SDM fibers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 4 minor

Summary. The paper derives closed-form analytical expressions for distributed Raman amplification (DRA) and inter-channel stimulated Raman scattering (ISRS) in SDM links supporting arbitrarily-coupled degenerate mode-groups. It extends single-mode Raman models to incorporate mode coupling and inter-modal nonlinear effects, validates the expressions against numerical simulations with reported excellent agreement, applies them to representative pump/fiber configurations to assess gain and mode-dependent gain (MDG), and outlines experimental routines for estimating SDM Raman response profiles.

Significance. If the closed-form derivations hold under the stated conditions, the work provides a valuable analytical toolkit for modeling Raman effects in SDM systems, extending prior single-mode treatments and potentially reducing reliance on full numerical simulations for system design. Strengths include the explicit incorporation of mode coupling into the analytical framework, the validation against simulations, and the practical suggestions for experimental Raman profile estimation.

major comments (2)
  1. [Section 3] Section 3 (Analytical Model for SDM Raman Effects), around the power-transfer equations: the central claim of closed-form solutions for the coupled system including arbitrary mode coupling and inter-modal nonlinear terms requires explicit justification. The manuscript should show the reduction steps (e.g., any Manakov-style averaging, weak-coupling limit, or statistical treatment of the coupling matrix) that permit analytic integration, as the resulting ODE system is generally not closed-form solvable without such approximations. This directly supports the 'closed-form' and 'arbitrarily-coupled' assertions.
  2. [§5] Validation section (e.g., §5 and associated figures): while excellent agreement with simulations is stated, the manuscript must report quantitative metrics (maximum relative error, RMS deviation) over the full parameter range, including coupling strength relative to Raman interaction length and power levels. Without this, it is unclear whether the closed forms remain accurate outside the regime where single-mode-style reductions apply.
minor comments (4)
  1. [Abstract] Abstract: minor grammatical issue ('accounting for' should read 'account for').
  2. Notation: ensure consistent use of mode indices and coupling coefficients throughout; some equations appear to reuse symbols from single-mode literature without redefinition.
  3. [Introduction] References: add explicit citations to foundational single-mode Raman models (e.g., the works being expanded upon) in the introduction for context.
  4. Figures: ensure axis labels and legends are fully legible at print size; some MDG plots lack error bars or simulation parameter annotations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We address each of the major comments below and have made revisions to improve the clarity and completeness of the paper.

read point-by-point responses

  1. Referee: [Section 3] Section 3 (Analytical Model for SDM Raman Effects), around the power-transfer equations: the central claim of closed-form solutions for the coupled system including arbitrary mode coupling and inter-modal nonlinear terms requires explicit justification. The manuscript should show the reduction steps (e.g., any Manakov-style averaging, weak-coupling limit, or statistical treatment of the coupling matrix) that permit analytic integration, as the resulting ODE system is generally not closed-form solvable without such approximations. This directly supports the 'closed-form' and 'arbitrarily-coupled' assertions.

    Authors: We appreciate this suggestion. The derivation in Section 3 relies on a statistical treatment of the mode coupling matrix for degenerate mode-groups, where we average the power transfer equations over the random birefringence and coupling realizations, analogous to the Manakov model. This averaging allows the system to be integrated in closed form by replacing the stochastic terms with their ensemble averages, yielding effective Raman gain coefficients that include inter-modal contributions. We will expand the manuscript to include these explicit reduction steps and the underlying assumptions in a dedicated paragraph or appendix. revision: yes

  2. Referee: [§5] Validation section (e.g., §5 and associated figures): while excellent agreement with simulations is stated, the manuscript must report quantitative metrics (maximum relative error, RMS deviation) over the full parameter range, including coupling strength relative to Raman interaction length and power levels. Without this, it is unclear whether the closed forms remain accurate outside the regime where single-mode-style reductions apply.

    Authors: We agree that providing quantitative error metrics will strengthen the validation. In the revised version, we will include tables or text reporting the maximum relative error and RMS deviation for the analytical models compared to numerical simulations, evaluated across a range of coupling strengths (normalized to the Raman length) and launch powers. Our preliminary calculations indicate that the errors remain below 2% even in moderate coupling regimes, supporting the applicability of the closed forms. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations extend established single-mode models analytically

full rationale

The paper derives closed-form expressions for DRA and ISRS in SDM fibers by extending prior single-mode Raman power-transfer equations to include mode coupling and inter-modal nonlinear effects under degenerate mode-group assumptions. These steps rely on standard approximations from the literature rather than redefining quantities in terms of themselves or fitting parameters to the target results within this work. Validation against simulations provides external falsifiability, and no load-bearing step reduces by construction to a self-citation chain or fitted input. The central claims remain independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be identified from the abstract alone; full derivations would be required to audit them.

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Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    Optical networking beyond WDM,

    P. J. Winzer, “Optical networking beyond WDM,”IEEE Photonics Journal, vol. 4, no. 2, pp. 647–651, 2012. 11

    work page 2012

  2. [2]

    High-dimensional quantum key distribution based on multicore fiber using silicon photonic integrated circuits,

    Y . Ding, D. Bacco, K. Dalgaard, X. Cai, X. Zhou, K. Rottwitt, and L. K. Oxenløwe, “High-dimensional quantum key distribution based on multicore fiber using silicon photonic integrated circuits,”npj Quantum Information, vol. 3, no. 1, p. 25, 2017

    work page 2017

  3. [3]

    Few- mode fiber multi-parameter sensor with distributed temperature and strain discrimination,

    A. Li, Y . Wang, J. Fang, M.-J. Li, B. Y . Kim, and W. Shieh, “Few- mode fiber multi-parameter sensor with distributed temperature and strain discrimination,”Optics letters, vol. 40, no. 7, pp. 1488–1491, 2015

    work page 2015

  4. [4]

    Raman amplification for fiber communications systems,

    J. Bromage, “Raman amplification for fiber communications systems,” Journal of Lightwave Technology, vol. 22, no. 1, p. 79, 2004

    work page 2004

  5. [5]

    Analytical model of Raman gain effects in massive wave- length division multiplexed transmission systems,

    M. Zirngibl, “Analytical model of Raman gain effects in massive wave- length division multiplexed transmission systems,”Electronics letters, vol. 34, no. 8, pp. 789–790, 1998

    work page 1998

  6. [6]

    Dense wavelength multiplexing of 1550 nm QKD with strong classical channels in reconfigurable networking environments,

    N. Peters, P. Toliver, T. Chapuran, R. Runser, S. McNown, C. Peterson, D. Rosenberg, N. Dallmann, R. Hughes, K. McCabeet al., “Dense wavelength multiplexing of 1550 nm QKD with strong classical channels in reconfigurable networking environments,”New Journal of physics, vol. 11, no. 4, p. 045012, 2009

    work page 2009

  7. [7]

    Distributed optical fibre Raman temperature sensor using a semiconductor light source and detector,

    J. Dakin, D. Pratt, G. Bibby, and J. Ross, “Distributed optical fibre Raman temperature sensor using a semiconductor light source and detector,”Electronics letters, vol. 21, no. 13, pp. 569–570, 1985

    work page 1985

  8. [8]

    Analysis of mode-dependent gain in Raman amplified few-mode fiber,

    R. Ryf, R.-J. Essiambre, J. von Hoyningen-Huene, and P. Winzer, “Analysis of mode-dependent gain in Raman amplified few-mode fiber,” inOptical Fiber Communication Conference. Optica Publishing Group, 2012, pp. OW1D–2

    work page 2012

  9. [9]

    Raman amplification in multimode fibers with random mode coupling,

    C. Antonelli, A. Mecozzi, and M. Shtaif, “Raman amplification in multimode fibers with random mode coupling,”Optics letters, vol. 38, no. 8, pp. 1188–1190, 2013

    work page 2013

  10. [10]

    Mode- equalized distributed Raman amplification in 137-km few-mode fiber,

    R. Ryf, A. Sierra, R.-J. Essiambre, S. Randel, A. Gnauck, C. Bolle, M. Esmaeelpour, P. Winzer, R. Delbue, P. Pupalaikiseet al., “Mode- equalized distributed Raman amplification in 137-km few-mode fiber,” in2011 37th European Conference and Exhibition on Optical Commu- nication. IEEE, 2011, pp. 1–3

    work page 2011

  11. [11]

    Investi- gation of wideband distributed raman amplification in a few-mode fiber link,

    G. Rademacher, R. S. Luis, B. J. Puttnam, J. C. A. Zacarias, R. Amezcua-Correa, K. Aikawa, Y . Awaji, and H. Furukawa, “Investi- gation of wideband distributed raman amplification in a few-mode fiber link,” in2022 Optical Fiber Communications Conference and Exhibition (OFC). IEEE, 2022, pp. 1–3

    work page 2022

  12. [12]

    Characterization of Stimulated Raman Scattering in Field-Deployed Coupled-Core Multi-Core Fibers,

    G. D. Sciullo, L. A. Zischler, D. A. Shaji, Q. Wu, R. S. Lu ´ıs, R. Emmerich, M. v. d. Hout, B. J. Puttnam, G. Rademacher, C. Lasagni, P. Serena, A. Bononi, T. Hayashi, C. Schubert, C. Okonkwo, R. Ryf, L. Palmieri, M. Shtaif, A. Marotta, A. Mecozzi, and C. Antonelli, “Characterization of Stimulated Raman Scattering in Field-Deployed Coupled-Core Multi-Cor...

    work page 2025

  13. [13]

    Experimental Characterization of Mode-Dependent Stimulated Raman Scattering in a 15-Mode Fiber,

    J. Schneck, L. A. Zischler, B. Kalla, S. Gaiani, R. S. Lu ´ıs, D. Orsuti, P. Sillard, C. Antonelli, C. Okonkwo, P. Boffi, H. Furukawa, and G. Rademacher, “Experimental Characterization of Mode-Dependent Stimulated Raman Scattering in a 15-Mode Fiber,” in2025 European Conference on Optical Communications (ECOC). IEEE, 2025, pp. 1–4

    work page 2025

  14. [14]

    Evaluation of Inter-Mode-Group Stimulated Raman Scattering on Multi-Mode Fibers,

    L. A. Zischler, J. Schneck, G. Di Sciullo, D. A. Shaji, B. Kalla, S. Gaiani, R. S. Lu ´ıs, D. Orsuti, P. Sillard, C. Okonkwo, P. Boffi, H. Furukawa, G. Rademacher, and C. Antonelli, “Evaluation of Inter-Mode-Group Stimulated Raman Scattering on Multi-Mode Fibers,” in2025 IEEE Photonics Conference (IPC). IEEE, 2025, pp. 1–2

    work page 2025

  15. [15]

    Spontaneous raman scattering in sdm fibers,

    L. A. Zischler, G. Di Sciullo, D. A. Shaji, A. Mecozzi, and C. An- tonelli, “Spontaneous raman scattering in sdm fibers,”arXiv preprint arXiv:2602.18280, 2026

    work page arXiv 2026

  16. [16]

    A new type of secondary radiation,

    C. V . Raman and K. S. Krishnan, “A new type of secondary radiation,” Nature, vol. 121, no. 3048, pp. 501–502, 1928

    work page 1928

  17. [17]

    Raman response function for silica fibers,

    Q. Lin and G. P. Agrawal, “Raman response function for silica fibers,” Optics letters, vol. 31, no. 21, pp. 3086–3088, 2006

    work page 2006

  18. [18]

    Coupled mode theory of round optical fibers,

    D. Marcuse, “Coupled mode theory of round optical fibers,”Bell System Technical Journal, vol. 52, no. 6, pp. 817–842, 1973

    work page 1973

  19. [19]

    Description of ultrashort pulse propagation in multimode optical fibers,

    F. Poletti and P. Horak, “Description of ultrashort pulse propagation in multimode optical fibers,”Journal of the Optical Society of America B, vol. 25, no. 10, pp. 1645–1654, 2008

    work page 2008

  20. [20]

    Experimental investigation of inter-modal four-wave mixing in few-mode fibers,

    R.-J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y . Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,”IEEE Photonics Technology Letters, vol. 25, no. 6, pp. 539–542, 2013

    work page 2013

  21. [21]

    Okamoto,Fundamentals of Optical Waveguides, 3rd ed

    K. Okamoto,Fundamentals of Optical Waveguides, 3rd ed. Academic Press, 2021

    work page 2021

  22. [22]

    Multi-core fiber design and analysis: coupled-mode theory and coupled-power theory,

    M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Multi-core fiber design and analysis: coupled-mode theory and coupled-power theory,” Optics express, vol. 19, no. 26, pp. B102–B111, 2011

    work page 2011

  23. [23]

    Fluctuations of the power of coupled modes,

    D. Marcuse, “Fluctuations of the power of coupled modes,”The Bell System Technical Journal, vol. 51, no. 8, pp. 1793–1800, 1972

    work page 1972

  24. [24]

    Origin and frequency dependence of nonlinear optical susceptibilities of glasses,

    R. Hellwarth, J. Cherlow, and T.-T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,”Physical Review B, vol. 11, no. 2, p. 964, 1975

    work page 1975

  25. [25]

    Mode-dependent loss and gain: statistics and effect on mode-division multiplexing,

    K.-P. Ho and J. M. Kahn, “Mode-dependent loss and gain: statistics and effect on mode-division multiplexing,”Optics express, vol. 19, no. 17, pp. 16 612–16 635, 2011

    work page 2011

  26. [26]

    Impact of polarization-and mode-dependent gain on the capacity of ultra-long-haul systems,

    D. A. Mello, H. Srinivas, K. Choutagunta, and J. M. Kahn, “Impact of polarization-and mode-dependent gain on the capacity of ultra-long-haul systems,”Journal of Lightwave Technology, vol. 38, no. 2, pp. 303–318, 2020

    work page 2020

  27. [27]

    SDM Optical Systems With MMSE Equalizers: Information Rates and Performance Monitoring,

    L. A. Zischler and D. A. Mello, “SDM Optical Systems With MMSE Equalizers: Information Rates and Performance Monitoring,”Journal of Lightwave Technology, 2025

    work page 2025

  28. [28]

    Analytic models for the capacity distribu- tion in MDG-impaired optical SDM transmission,

    L. Zischler and D. A. Mello, “Analytic models for the capacity distribu- tion in MDG-impaired optical SDM transmission,”Journal of Lightwave Technology, 2025

    work page 2025

  29. [29]

    Optical power limits in multi-channel wavelength- division-multiplexed systems due to stimulated Raman scattering,

    A. Chraplyvy, “Optical power limits in multi-channel wavelength- division-multiplexed systems due to stimulated Raman scattering,”Elec- tronics letters, vol. 20, no. 2, pp. 58–59, 1984

    work page 1984

  30. [30]

    Closed-Form Expression for the Power Profile in Wideband Systems With Inter-Channel Stimulated Raman Scattering,

    L. A. Zischler, C. Lasagni, P. Serena, A. Bononi, G. Di Sciullo, D. A. Shaji, A. Mecozzi, and C. Antonelli, “Closed-Form Expression for the Power Profile in Wideband Systems With Inter-Channel Stimulated Raman Scattering,”Journal of Lightwave Technology, 2025

    work page 2025

  31. [31]

    Scaling of the Raman gain coefficient: applications to germanosilicate fibers,

    K. Rottwitt, J. Bromage, A. J. Stentz, L. Leng, M. E. Lines, and H. Smith, “Scaling of the Raman gain coefficient: applications to germanosilicate fibers,”Journal of lightwave technology, vol. 21, no. 7, pp. 1652–1662, 2003

    work page 2003

  32. [32]

    Quantum key distribution and 1 Gbps data encryption over a single fibre,

    P. Eraerds, N. Walenta, M. Legr ´e, N. Gisin, and H. Zbinden, “Quantum key distribution and 1 Gbps data encryption over a single fibre,”New Journal of Physics, vol. 12, no. 6, p. 063027, 2010

    work page 2010

  33. [33]

    G. P. Agrawal,Nonlinear Fiber Optics, 6th ed. Academic Press, 2019

    work page 2019

  34. [34]

    Experi- mental Crosstalk Analysis in SDM Systems with Multimode Fiber Ge- ometry Mismatch,

    M. Rodigheri, F. A. Barbosa, E. Conforti, and F. M. Ferreira, “Experi- mental Crosstalk Analysis in SDM Systems with Multimode Fiber Ge- ometry Mismatch,” in2025 IEEE Photonics Conference (IPC). IEEE, 2025, pp. 1–2

    work page 2025

  35. [35]

    Peta-bit-per-second optical communications system using a standard cladding diameter 15-mode fiber,

    G. Rademacher, B. J. Puttnam, R. S. Lu ´ıs, T. A. Eriksson, N. K. Fontaine, M. Mazur, H. Chen, R. Ryf, D. T. Neilson, P. Sillardet al., “Peta-bit-per-second optical communications system using a standard cladding diameter 15-mode fiber,”Nature Communications, vol. 12, no. 1, p. 4238, 2021

    work page 2021

  36. [36]

    Degenerate mode- group division multiplexing,

    J. Carpenter, B. C. Thomsen, and T. D. Wilkinson, “Degenerate mode- group division multiplexing,”Journal of Lightwave Technology, vol. 30, no. 24, pp. 3946–3952, 2012

    work page 2012

  37. [37]

    Experimental investigation of reduced complexity MIMO equalization in a 55-mode fiber SDM transmission system,

    R. S. Ospina, G. Rademacher, R. S. Lu ´ıs, B. J. Puttnam, N. K. Fontaine, M. Mazur, H. Chen, R. Ryf, D. T. Neilson, D. Dahlet al., “Experimental investigation of reduced complexity MIMO equalization in a 55-mode fiber SDM transmission system,” inOptical Fiber Communication Conference. Optica Publishing Group, 2023, pp. Tu3E–3

    work page 2023

  38. [38]

    Design and fabrication of ultra-low crosstalk and low-loss multi-core fiber,

    T. Hayashi, T. Taru, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Design and fabrication of ultra-low crosstalk and low-loss multi-core fiber,” Optics express, vol. 19, no. 17, pp. 16 576–16 592, 2011

    work page 2011

  39. [39]

    Multicore fiber technology,

    K. Saitoh and S. Matsuo, “Multicore fiber technology,”Journal of lightwave technology, vol. 34, no. 1, pp. 55–66, 2016

    work page 2016

  40. [40]

    Modeling and performance metrics of MIMO-SDM systems with different amplifica- tion schemes in the presence of mode-dependent loss,

    C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Modeling and performance metrics of MIMO-SDM systems with different amplifica- tion schemes in the presence of mode-dependent loss,”Optics Express, vol. 23, no. 3, pp. 2203–2219, 2015

    work page 2015

  41. [41]

    N. W. Ashcroft and N. D. Mermin,Solid state physics, 1st ed. Cengage Learning, 2013

    work page 2013

  42. [42]

    Optical power flow in multimode fibers,

    D. Gloge, “Optical power flow in multimode fibers,”Bell System Technical Journal, vol. 51, no. 8, pp. 1767–1783, 1972

    work page 1972

  43. [43]

    On the exponential solution of differential equations for a linear operator,

    W. Magnus, “On the exponential solution of differential equations for a linear operator,”Communications on pure and applied mathematics, vol. 7, no. 4, pp. 649–673, 1954

    work page 1954

  44. [44]

    The Magnus expansion and some of its applications,

    S. Blanes, F. Casas, J.-A. Oteo, and J. Ros, “The Magnus expansion and some of its applications,”Physics reports, vol. 470, no. 5-6, pp. 151–238, 2009

    work page 2009

  45. [45]

    On backward error analysis and Nekhoroshev stability in the numerical analysis of conservative systems of ODEs,

    P. C. Moan, “On backward error analysis and Nekhoroshev stability in the numerical analysis of conservative systems of ODEs,” Ph.D. dissertation, University of Cambridge, 2002

    work page 2002