Mathematical Modeling of Electromagnetic Wave Propagation and Non-linear Effects in Single-mode Optical Fibers

Randriana Heritiana Nambinina Erica; Ando Nirina Andriamanalina

doi:doi:10.11648/j.awcn.20261101.12

Methodology Article |

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Mathematical Modeling of Electromagnetic Wave Propagation and Non-linear Effects in Single-mode Optical Fibers

Randriana Heritiana Nambinina Erica^*

, Ando Nirina Andriamanalina

Published in Advances in Wireless Communications and Networks (Volume 11, Issue 1)

Received: 9 October 2026 Accepted: 17 October 2026 Published: 28 October 2026

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Abstract

Modern high-capacity optical fiber communication systems rely fundamentally on accurate mathematical models capable of describing the complex mechanisms governing optical pulse propagation in dielectric waveguides. As transmission rates and propagation distances continue to increase, the combined effects of chromatic dispersion and optical nonlinearities become increasingly significant and must be carefully considered in the design and optimization of advanced communication links. This article presents a unified mathematical framework for modeling light propagation in single-mode optical fibers, starting from Maxwell’s macroscopic equations and progressively deriving the wave equation, the slowly varying envelope approximation, the Nonlinear Schrödinger Equation (NLSE), and its generalized form, the Generalized Nonlinear Schrödinger Equation (GNLSE). The formulation provides a consistent description of the main linear and nonlinear phenomena affecting optical signals during propagation. Particular attention is devoted to chromatic dispersion, group velocity dispersion (GVD), higher-order dispersion effects, fiber attenuation, and Kerr-induced nonlinear interactions, including Self-Phase Modulation (SPM), Cross-Phase Modulation (XPM), and Four-Wave Mixing (FWM). The physical origin and mathematical representation of these mechanisms are examined to clarify their influence on pulse shape, temporal broadening, spectral evolution, and transmission performance. Numerical simulations based on the Split-Step Fourier Method (SSFM) are implemented in Python to solve the propagation equations efficiently and investigate the evolution of optical pulses along standard SMF-28 transmission links. The simulations illustrate characteristic effects such as dispersion-induced pulse broadening, nonlinear frequency chirping, spectral modification, and the formation and propagation of optical solitons resulting from the balance between anomalous dispersion and Kerr nonlinearity. The impact of these propagation effects on system performance is also evaluated through bit error rate (BER) degradation and signal distortion. By combining theoretical derivations, numerical modeling, and physical interpretation, this study establishes a comprehensive framework connecting fundamental electromagnetic theory with practical fiber-optic transmission analysis. The proposed approach provides useful guidelines for understanding propagation limitations, optimizing system parameters, and developing next-generation high-capacity optical communication networks, particularly for terabit-scale and ultra-long-haul transmission systems in which dispersion and nonlinear effects represent major constraints on achievable performance.

Published in	Advances in Wireless Communications and Networks (Volume 11, Issue 1)
DOI	10.11648/j.awcn.20261101.12
Page(s)	7-17
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Optical Fiber Propagation, Maxwell Equations, Non-linear Schrödinger Equation, Chromatic Dispersion, Self-phase Modulation, Optical Solitons, Split-step Fourier Method

References

[1]	G. P. Agrawal, Fiber-Optic Communication Systems, 5th ed. Hoboken, NJ, USA: John Wiley & Sons, 2021. https://doi.org/10.1002/9781119737391
[2]	G. P. Agrawal, Nonlinear Fiber Optics, 6th ed. San Diego, CA, USA: Academic Press, 2019. https://doi.org/10.1016/C2018-0-01168-8
[3]	J. D. Jackson, Classical Electrodynamics, 3rd ed. New York, NY, USA: John Wiley & Sons, 1999. DOI not available.
[4]	G. Keiser, Optical Fiber Communications, 4th ed. New York, NY, USA: McGraw-Hill, 2010. DOI not available.
[5]	A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. Oxford, UK: Oxford University Press, 2007. DOI not available.
[6]	R. W. Boyd, Nonlinear Optics, 4th ed. Burlington, MA, USA: Academic Press, 2020. https://doi.org/10.1016/C2015-0-05510-1
[7]	L. F. Mollenauer and J. P. Gordon, Solitons in Optical Fibers: Fundamentals and Applications. San Diego, CA, USA: Academic Press, 2006. https://doi.org/10.1016/B978-0-12-504190-4.X5000-2
[8]	C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE Journal of Quantum Electronics, vol. 23, no. 2, pp. 174-176, Feb. 1987. https://doi.org/10.1109/JQE.1987.1073308
[9]	E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” Journal of Lightwave Technology, vol. 26, no. 20, pp. 3416-3425, Oct. 2008. https://doi.org/10.1109/JLT.2008.927791
[10]	D. Zibar, M. Piels, R. Jones, and C. G. Schäeffer, “Machine learning techniques in optical communication,” Journal of Lightwave Technology, vol. 34, no. 6, pp. 1442-1452, Mar. 2016. https://doi.org/10.1109/JLT.2015.2508502
[11]	T. Erdogan, “Fiber grating spectra,” Journal of Lightwave Technology, vol. 15, no. 8, pp. 1277-1294, 1997. https://doi.org/10.1109/50.618322
[12]	C. R. Pollock and M. Lipson, Integrated Photonics. New York, NY, USA: Springer, 2003. https://doi.org/10.1007/978-1-4757-5522-0
[13]	R. Ramaswami, K. N. Sivarajan, and G. H. Sasaki, Optical Networks: A Practical Perspective, 3rd ed. Burlington, MA, USA: Morgan Kaufmann, 2009. DOI not confirmed for the 3rd edition.
[14]	X. Bao and L. Chen, “Recent progress in fiber Bragg grating sensors,” Sensors, vol. 12, no. 7, pp. 9267-9309, 2012. https://doi.org/10.3390/s120709267
[15]	L. N. Binh, Optical Fiber Communication Systems with MATLAB and Simulink Models, 2nd ed. Boca Raton, FL, USA: CRC Press, 2015. https://doi.org/10.1201/b17781

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Erica, R. H. N., Andriamanalina, A. N. (2026). Mathematical Modeling of Electromagnetic Wave Propagation and Non-linear Effects in Single-mode Optical Fibers. Advances in Wireless Communications and Networks, 11(1), 7-17. https://doi.org/10.11648/j.awcn.20261101.12

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ACS Style

Erica, R. H. N.; Andriamanalina, A. N. Mathematical Modeling of Electromagnetic Wave Propagation and Non-linear Effects in Single-mode Optical Fibers. Adv. Wirel. Commun. Netw. 2026, 11(1), 7-17. doi: 10.11648/j.awcn.20261101.12

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AMA Style

Erica RHN, Andriamanalina AN. Mathematical Modeling of Electromagnetic Wave Propagation and Non-linear Effects in Single-mode Optical Fibers. Adv Wirel Commun Netw. 2026;11(1):7-17. doi: 10.11648/j.awcn.20261101.12

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@article{10.11648/j.awcn.20261101.12,
  author = {Randriana Heritiana Nambinina Erica and Ando Nirina Andriamanalina},
  title = {Mathematical Modeling of Electromagnetic Wave Propagation and Non-linear Effects in Single-mode Optical Fibers},
  journal = {Advances in Wireless Communications and Networks},
  volume = {11},
  number = {1},
  pages = {7-17},
  doi = {10.11648/j.awcn.20261101.12},
  url = {https://doi.org/10.11648/j.awcn.20261101.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.awcn.20261101.12},
  abstract = {Modern high-capacity optical fiber communication systems rely fundamentally on accurate mathematical models capable of describing the complex mechanisms governing optical pulse propagation in dielectric waveguides. As transmission rates and propagation distances continue to increase, the combined effects of chromatic dispersion and optical nonlinearities become increasingly significant and must be carefully considered in the design and optimization of advanced communication links. This article presents a unified mathematical framework for modeling light propagation in single-mode optical fibers, starting from Maxwell’s macroscopic equations and progressively deriving the wave equation, the slowly varying envelope approximation, the Nonlinear Schrödinger Equation (NLSE), and its generalized form, the Generalized Nonlinear Schrödinger Equation (GNLSE). The formulation provides a consistent description of the main linear and nonlinear phenomena affecting optical signals during propagation. Particular attention is devoted to chromatic dispersion, group velocity dispersion (GVD), higher-order dispersion effects, fiber attenuation, and Kerr-induced nonlinear interactions, including Self-Phase Modulation (SPM), Cross-Phase Modulation (XPM), and Four-Wave Mixing (FWM). The physical origin and mathematical representation of these mechanisms are examined to clarify their influence on pulse shape, temporal broadening, spectral evolution, and transmission performance. Numerical simulations based on the Split-Step Fourier Method (SSFM) are implemented in Python to solve the propagation equations efficiently and investigate the evolution of optical pulses along standard SMF-28 transmission links. The simulations illustrate characteristic effects such as dispersion-induced pulse broadening, nonlinear frequency chirping, spectral modification, and the formation and propagation of optical solitons resulting from the balance between anomalous dispersion and Kerr nonlinearity. The impact of these propagation effects on system performance is also evaluated through bit error rate (BER) degradation and signal distortion. By combining theoretical derivations, numerical modeling, and physical interpretation, this study establishes a comprehensive framework connecting fundamental electromagnetic theory with practical fiber-optic transmission analysis. The proposed approach provides useful guidelines for understanding propagation limitations, optimizing system parameters, and developing next-generation high-capacity optical communication networks, particularly for terabit-scale and ultra-long-haul transmission systems in which dispersion and nonlinear effects represent major constraints on achievable performance.},
 year = {2026}
}

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TY  - JOUR
T1  - Mathematical Modeling of Electromagnetic Wave Propagation and Non-linear Effects in Single-mode Optical Fibers
AU  - Randriana Heritiana Nambinina Erica
AU  - Ando Nirina Andriamanalina
Y1  - 2026/10/28
PY  - 2026
N1  - https://doi.org/10.11648/j.awcn.20261101.12
DO  - 10.11648/j.awcn.20261101.12
T2  - Advances in Wireless Communications and Networks
JF  - Advances in Wireless Communications and Networks
JO  - Advances in Wireless Communications and Networks
SP  - 7
EP  - 17
PB  - Science Publishing Group
SN  - 2575-596X
UR  - https://doi.org/10.11648/j.awcn.20261101.12
AB  - Modern high-capacity optical fiber communication systems rely fundamentally on accurate mathematical models capable of describing the complex mechanisms governing optical pulse propagation in dielectric waveguides. As transmission rates and propagation distances continue to increase, the combined effects of chromatic dispersion and optical nonlinearities become increasingly significant and must be carefully considered in the design and optimization of advanced communication links. This article presents a unified mathematical framework for modeling light propagation in single-mode optical fibers, starting from Maxwell’s macroscopic equations and progressively deriving the wave equation, the slowly varying envelope approximation, the Nonlinear Schrödinger Equation (NLSE), and its generalized form, the Generalized Nonlinear Schrödinger Equation (GNLSE). The formulation provides a consistent description of the main linear and nonlinear phenomena affecting optical signals during propagation. Particular attention is devoted to chromatic dispersion, group velocity dispersion (GVD), higher-order dispersion effects, fiber attenuation, and Kerr-induced nonlinear interactions, including Self-Phase Modulation (SPM), Cross-Phase Modulation (XPM), and Four-Wave Mixing (FWM). The physical origin and mathematical representation of these mechanisms are examined to clarify their influence on pulse shape, temporal broadening, spectral evolution, and transmission performance. Numerical simulations based on the Split-Step Fourier Method (SSFM) are implemented in Python to solve the propagation equations efficiently and investigate the evolution of optical pulses along standard SMF-28 transmission links. The simulations illustrate characteristic effects such as dispersion-induced pulse broadening, nonlinear frequency chirping, spectral modification, and the formation and propagation of optical solitons resulting from the balance between anomalous dispersion and Kerr nonlinearity. The impact of these propagation effects on system performance is also evaluated through bit error rate (BER) degradation and signal distortion. By combining theoretical derivations, numerical modeling, and physical interpretation, this study establishes a comprehensive framework connecting fundamental electromagnetic theory with practical fiber-optic transmission analysis. The proposed approach provides useful guidelines for understanding propagation limitations, optimizing system parameters, and developing next-generation high-capacity optical communication networks, particularly for terabit-scale and ultra-long-haul transmission systems in which dispersion and nonlinear effects represent major constraints on achievable performance.
VL  - 11
IS  - 1
ER  -

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Author Information

Randriana Heritiana Nambinina Erica

Telecommunications, Doctoral School of Engineering Sciences and Innovation, Antananarivo, Madagascar

Contact Email

http://orcid.org/0009-0003-8089-172X
Ando Nirina Andriamanalina

Telecommunications, Doctoral School of Engineering Sciences and Innovation, Antananarivo, Madagascar

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Plain Text BibTeX RIS

APA Style

Erica, R. H. N., Andriamanalina, A. N. (2026). Mathematical Modeling of Electromagnetic Wave Propagation and Non-linear Effects in Single-mode Optical Fibers. Advances in Wireless Communications and Networks, 11(1), 7-17. https://doi.org/10.11648/j.awcn.20261101.12

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ACS Style

Erica, R. H. N.; Andriamanalina, A. N. Mathematical Modeling of Electromagnetic Wave Propagation and Non-linear Effects in Single-mode Optical Fibers. Adv. Wirel. Commun. Netw. 2026, 11(1), 7-17. doi: 10.11648/j.awcn.20261101.12

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AMA Style

Erica RHN, Andriamanalina AN. Mathematical Modeling of Electromagnetic Wave Propagation and Non-linear Effects in Single-mode Optical Fibers. Adv Wirel Commun Netw. 2026;11(1):7-17. doi: 10.11648/j.awcn.20261101.12

Copy | Download

@article{10.11648/j.awcn.20261101.12,
  author = {Randriana Heritiana Nambinina Erica and Ando Nirina Andriamanalina},
  title = {Mathematical Modeling of Electromagnetic Wave Propagation and Non-linear Effects in Single-mode Optical Fibers},
  journal = {Advances in Wireless Communications and Networks},
  volume = {11},
  number = {1},
  pages = {7-17},
  doi = {10.11648/j.awcn.20261101.12},
  url = {https://doi.org/10.11648/j.awcn.20261101.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.awcn.20261101.12},
  abstract = {Modern high-capacity optical fiber communication systems rely fundamentally on accurate mathematical models capable of describing the complex mechanisms governing optical pulse propagation in dielectric waveguides. As transmission rates and propagation distances continue to increase, the combined effects of chromatic dispersion and optical nonlinearities become increasingly significant and must be carefully considered in the design and optimization of advanced communication links. This article presents a unified mathematical framework for modeling light propagation in single-mode optical fibers, starting from Maxwell’s macroscopic equations and progressively deriving the wave equation, the slowly varying envelope approximation, the Nonlinear Schrödinger Equation (NLSE), and its generalized form, the Generalized Nonlinear Schrödinger Equation (GNLSE). The formulation provides a consistent description of the main linear and nonlinear phenomena affecting optical signals during propagation. Particular attention is devoted to chromatic dispersion, group velocity dispersion (GVD), higher-order dispersion effects, fiber attenuation, and Kerr-induced nonlinear interactions, including Self-Phase Modulation (SPM), Cross-Phase Modulation (XPM), and Four-Wave Mixing (FWM). The physical origin and mathematical representation of these mechanisms are examined to clarify their influence on pulse shape, temporal broadening, spectral evolution, and transmission performance. Numerical simulations based on the Split-Step Fourier Method (SSFM) are implemented in Python to solve the propagation equations efficiently and investigate the evolution of optical pulses along standard SMF-28 transmission links. The simulations illustrate characteristic effects such as dispersion-induced pulse broadening, nonlinear frequency chirping, spectral modification, and the formation and propagation of optical solitons resulting from the balance between anomalous dispersion and Kerr nonlinearity. The impact of these propagation effects on system performance is also evaluated through bit error rate (BER) degradation and signal distortion. By combining theoretical derivations, numerical modeling, and physical interpretation, this study establishes a comprehensive framework connecting fundamental electromagnetic theory with practical fiber-optic transmission analysis. The proposed approach provides useful guidelines for understanding propagation limitations, optimizing system parameters, and developing next-generation high-capacity optical communication networks, particularly for terabit-scale and ultra-long-haul transmission systems in which dispersion and nonlinear effects represent major constraints on achievable performance.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Mathematical Modeling of Electromagnetic Wave Propagation and Non-linear Effects in Single-mode Optical Fibers
AU  - Randriana Heritiana Nambinina Erica
AU  - Ando Nirina Andriamanalina
Y1  - 2026/10/28
PY  - 2026
N1  - https://doi.org/10.11648/j.awcn.20261101.12
DO  - 10.11648/j.awcn.20261101.12
T2  - Advances in Wireless Communications and Networks
JF  - Advances in Wireless Communications and Networks
JO  - Advances in Wireless Communications and Networks
SP  - 7
EP  - 17
PB  - Science Publishing Group
SN  - 2575-596X
UR  - https://doi.org/10.11648/j.awcn.20261101.12
AB  - Modern high-capacity optical fiber communication systems rely fundamentally on accurate mathematical models capable of describing the complex mechanisms governing optical pulse propagation in dielectric waveguides. As transmission rates and propagation distances continue to increase, the combined effects of chromatic dispersion and optical nonlinearities become increasingly significant and must be carefully considered in the design and optimization of advanced communication links. This article presents a unified mathematical framework for modeling light propagation in single-mode optical fibers, starting from Maxwell’s macroscopic equations and progressively deriving the wave equation, the slowly varying envelope approximation, the Nonlinear Schrödinger Equation (NLSE), and its generalized form, the Generalized Nonlinear Schrödinger Equation (GNLSE). The formulation provides a consistent description of the main linear and nonlinear phenomena affecting optical signals during propagation. Particular attention is devoted to chromatic dispersion, group velocity dispersion (GVD), higher-order dispersion effects, fiber attenuation, and Kerr-induced nonlinear interactions, including Self-Phase Modulation (SPM), Cross-Phase Modulation (XPM), and Four-Wave Mixing (FWM). The physical origin and mathematical representation of these mechanisms are examined to clarify their influence on pulse shape, temporal broadening, spectral evolution, and transmission performance. Numerical simulations based on the Split-Step Fourier Method (SSFM) are implemented in Python to solve the propagation equations efficiently and investigate the evolution of optical pulses along standard SMF-28 transmission links. The simulations illustrate characteristic effects such as dispersion-induced pulse broadening, nonlinear frequency chirping, spectral modification, and the formation and propagation of optical solitons resulting from the balance between anomalous dispersion and Kerr nonlinearity. The impact of these propagation effects on system performance is also evaluated through bit error rate (BER) degradation and signal distortion. By combining theoretical derivations, numerical modeling, and physical interpretation, this study establishes a comprehensive framework connecting fundamental electromagnetic theory with practical fiber-optic transmission analysis. The proposed approach provides useful guidelines for understanding propagation limitations, optimizing system parameters, and developing next-generation high-capacity optical communication networks, particularly for terabit-scale and ultra-long-haul transmission systems in which dispersion and nonlinear effects represent major constraints on achievable performance.
VL  - 11
IS  - 1
ER  -

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