Fast Numerical Nonlinear Fourier Transform Algorithms for the Manakov Equation

Master thesis (2021)

Authors

L. de Vries Mechanical Engineering

Contributors

S. Chimmalgi Team Sander Wahls - Mechanical, Maritime and Materials Engineering (mentor)
S. Wahls Team Sander Wahls - Mechanical, Maritime and Materials Engineering (mentor)
K. Batselier Team Kim Batselier - Mechanical, Maritime and Materials Engineering (graduation committee member)
Faculty
Mechanical Engineering, Mechanical Engineering
Copyright: © 2021 Lianne de Vries
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Published Date

15-10-2021

Language

English

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Faculty

Mechanical Engineering

Abstract

Optical fibers form the backbone of our global data transmission infrastructure. As demands on global data transmission grow the capacity of these systems needs to be increased. The behaviour of light waves through these optical fibers is described by the Manakov Equation (ME), a system of nonlinear partial differential equations.

The ME is an integrable system, which can be solved analytically using Nonlinear Fourier Transforms. Recently, fiber-optic communication systems based on the Nonlinear Fourier Transform (NFT) of the ME have been proposed. Similar to the linear Fourier Transform, which decomposes a signal in linear frequency components, the NFT decomposes a signal in nonlinear frequency components. This nonlinear spectrum consists of a continuous and a discrete part. The continuous spectrum in general constitutes the whole real line. The discrete spectrum consists of distinct points in the complex plane which correspond to so-called solitons, which are stable wave forms. The evolution of the nonlinear spectrum along the fiber is trivial.

The nonlinear spectrum however cannot be computed analytically for most signals and therefore numerical methods are needed. The existing numerical methods have a high computational complexity of O(D^2) for computing the continuous spectrum, with D the number of time samples of the signal. For the Nonlinear Schrödinger Equation (NSE), a simplification of the ME, more efficient numerical methods exist with a computational complexity of O(D log^2(D)). In this thesis we present an extension of these so-called fast NFT methods to the ME. The resulting algorithms are second and fourth-order algorithms based on second and fourth order exponential integration methods respectively.

We developed open source software implementing the fast NFT algorithms for the ME and integrated them in the already existing Fast Nonlinear Fourier Transform (FNFT) software library. We provide detailed documentation and examples which allow other researchers to use the algorithms as tools or as a base for developing new algorithms. We furthermore test the accuracy of the developed algorithms against analytic examples. Of these examples, the rectangle signal and secant hyperbolic signal are new analytic examples for the ME to the best of our knowledge.