Abstract
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.
Highlights
This article is a continuation of our previous work [1], where we suggested a novel solution of coupled nonlinear Schrödinger equation system (CNSES) for simulation of ultra-short optical pulse propagation in birefringent fibers
The definitions are used in Equations (1) and (2): Wi —the complex envelope of the optical wave of the i-th mode, α(i) —attenuation of the i-th mode; β(i) 1, β(i) 2, β(i) 3 —the first, second and third order dispersion of the i-th mode respectively; γ(i) —parameter of nonlinearity for the i-th mode; Ci,m, Bi,m —coupling coefficients between the i-th and m-th modes; TR —Raman scattering parameter; ω(i) 0 —angular frequency of the i-th mode; z—coordinate along an optical fiber; t—time, j—imaginary one, TN —final time, and f (i) (t)—
The complex frequency characteristic is proportional to the non-zero function in the frequency channel in case of phase incursion described by Equation (18) without third-order dispersion (φ000 ($,L) = 0): H ( j ($ + Ω ), L )
Summary
Airat Zh. Sakhabutdinov 1, * , Vladimir I. JSC “Scientific Production Association State Optical Institute Named after Vavilov S.I.”, 36/1, Babushkin street, 192171 Saint Petersburg, Russia
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