Numerical Method for Coupled Nonlinear Schrödinger Equations in Few-Mode Fiber

This paper discusses novel approaches to the numerical integration of the coupled nonlinear Schrödinger equations system for few-mode wave propagation. The wave propagation assumes the propagation of up to nine modes of light in an optical fiber. In this case, the light propagation is described by the non-linear coupled Schrödinger equation system, where propagation of each mode is described by own Schrödinger equation with other modes’ interactions. In this case, the coupled nonlinear Schrödinger equation system (CNSES) solving becomes increasingly complex, because each mode affects the propagation of other modes. The suggested solution is based on the direct numerical integration approach, which is based on a finite-difference integration scheme. The well-known explicit finite-difference integration scheme approach fails due to the non-stability of the computing scheme. Owing to this, here we use the combined explicit/implicit finite-difference integration scheme, which is based on the implicit Crank–Nicolson finite-difference scheme. It ensures the stability of the computing scheme. Moreover, this approach allows separating the whole equation system on the independent equation system for each wave mode at each integration step. Additionally, the algorithm of numerical solution refining at each step and the integration method with automatic integration step selection are used. The suggested approach has a higher performance (resolution)—up to three times or more in comparison with the split-step Fourier method—since there is no need to produce direct and inverse Fourier transforms at each integration step. The key advantage of the developed approach is the calculation of any number of modes propagated in the fiber.

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Numerical Method for Coupled Nonlinear Schrödinger Equations in Few Mode Fiber

10.20944/preprints202012.0192.v1 ◽

2020 ◽

Author(s):

Airat Sakhabutdinov ◽

Vladimir Anfinogentov ◽

Oleg Morozov ◽

Vladimir Burdin ◽

Anton Bourdine ◽

...

Keyword(s):

Wave Propagation ◽

Finite Difference ◽

Schrödinger Equation ◽

Numerical Integration ◽

Schrodinger Equation ◽

Equation System ◽

Integration Scheme ◽

Integration Step ◽

Non Linear ◽

Computing Scheme

This paper discusses approaches to the numerical integration of the coupled nonlinear Schrödinger equations system in case of few-mode wave propagation. The wave propagation assumes the propagation of up to nine modes of light in an optical fiber. In this case, the light propagation is described by the non-linear coupled Schrödinger equation system, where propagation of each mode is described by own Schrödinger equation with other modes interactions. In this case, the non-linear coupled Schrödinger equation system solving becomes increasingly complex, because each mode affects the propagation of other modes. The suggested solution is based on the direct numerical integration approach, which is based on a finite-difference integration scheme. The well-known explicit finite-difference integration scheme approach fails, due to the non-stability of the computing scheme. Due to this fact, the combined explicit/implicit finite-difference integration scheme, based on the implicit Crank–Nicolson finite-difference scheme, is used. It allows ensuring the stability of the computing scheme. Moreover, this approach allows separating the whole equation system on the independent equation system for each wave mode at each integration step. Additionally, the algorithm of numerical solution refining at each step and the integration method with automatic integration step selection are used. The suggested approach has performance gains (or resolutions) up to three or more orders of magnitude in comparison with the split-step Fourier method due to the fact that there is no need to produce direct and inverse Fourier transforms at each integration step. The main advantage of the proposed method is the ability to calculate the propagation of an arbitrary number of modes in the fiber.

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Численные подходы к решению нелинейной системы уравнений Шрёдингера для распространения волн в оптическом волокне

Вычислительные технологии ◽

10.25743/ict.2020.25.2.007 ◽

2020 ◽

Author(s):

A.J. Sakhabutdinov ◽

V.I. Anfinogentov ◽

O.G. Morozov ◽

R.R. Gubaidullin

Keyword(s):

Finite Difference ◽

Nonlinear Term ◽

Initial Boundary ◽

Equation System ◽

Integration Scheme ◽

Integration Step ◽

Disturbance Propagation ◽

Calculation Results ◽

Definition Of ◽

Selection Of

The paper discusses approaches to the numerical integration of the second-kind Manakov equation system. Emphasis is placed on the transition from writing equations in dimensional quantities to equations in dimensionless units. A combined explicit/implicit finite-difference integration scheme based on the implicit CrankNicolson finite-difference scheme is proposed and substantiated, which allows integrating a nonlinear system of equations with a choice of nonlinear term at the previous integration step. An algorithm for leveling the disadvantage associated with the definition of the nonlinear term from the previous integration step is proposed. The approach of automatic selection of the integration step, which reduces the total number of integration steps while maintaining the required accuracy of the approximate solution, is substantiated. Examples of the calculation results for some values of the disturbance propagation are given. The limitations imposed by the computing scheme on the length of the integrable fiber section are described, and approaches, that eliminate these limitations without the need to increase arrays dimensions, are proposed. Requirements for initial boundary conditions are discussed. Предложена разработка метода приближенного решения системы уравнения Манакова как одного из частных случаев системы уравнений Шрёдингера, связанного с моделированием оптических линий связи на основе многомодовых волокон. Решение ищется методами численного интегрирования. Показано, что численное интегрирование может быть осуществлено с использованием комбинированной явно-неявной схемы численного интегрирования на основе схемы КранкаНиколсон с записью нелинейного слагаемого в конечно-разностной форме, взятого с предыдущего шага интегрирования. Использован алгоритм автоматического выбора шага интегрирования, реализован итерационный алгоритм уточнения решения на каждом шаге, предложен алгоритм, позволяющий производить расчет параметров на протяженных участках. Нахождение приближенного решения системы уравнения Манакова может быть осуществлено с использованием комбинированной явно-неявной схемы КранкаНиколсон, а запись нелинейного слагаемого в конечно-разностной форме, взятого с предыдущего шага интегрирования, дает неплохой результат. Алгоритм автоматического выбора шага интегрирования обеспечивает лучшую сходимость результатов интегрирования на большом расстоянии и снижение необходимого количества шагов интегрирования. Алгоритм уточнения решения на каждом шаге позволяет нивелировать недостаток метода явной записи неявного слагаемого и интегрировать с большим шагом. Алгоритм расчета параметров распространения возмущения со сдвигом фрейма позволяет сделать вывод о целесообразности развития этого алгоритма.

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Original Solution of Coupled Nonlinear Schrödinger Equations for Simulation of Ultrashort Optical Pulse Propagation in a Birefringent Fiber

Fibers ◽

10.3390/fib8060034 ◽

2020 ◽

Vol 8 (6) ◽

pp. 34 ◽

Cited By ~ 1

Author(s):

Airat Zhavdatovich Sakhabutdinov ◽

Vladimir Ivanovich Anfinogentov ◽

Oleg Gennadievich Morozov ◽

Vladimir Alexandrovich Burdin ◽

Anton Vladimirovich Bourdine ◽

...

Keyword(s):

Finite Difference ◽

Nonlinear Schrödinger Equations ◽

Integration Scheme ◽

Integration Step ◽

Schrödinger Equations ◽

Coupled Nonlinear Schrödinger Equations ◽

Physical Processes ◽

Nonlinear Schrödinger ◽

Nonlinear Schrodinger Equations ◽

Implicit Finite Difference

This paper discusses approaches to the numerical integration of the coupled nonlinear Schrödinger equations system, different from the generally accepted approach based on the method of splitting according to physical processes. A combined explicit/implicit finite-difference integration scheme based on the implicit Crank–Nicolson finite-difference scheme is proposed and substantiated. It allows the integration of a nonlinear system of equations with a choice of nonlinear terms from the previous integration step. The main advantages of the proposed method are: its absolute stability through the use of an implicit finite-difference integration scheme and an integrated mechanism for refining the numerical solution at each step; integration with automatic step selection; performance gains (or resolutions) up to three or more orders of magnitude due to the fact that there is no need to produce direct and inverse Fourier transforms at each integration step, as is required in the method of splitting according to physical processes. An additional advantage of the proposed method is the ability to calculate the interaction with an arbitrary number of propagation modes in the fiber.

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