Embedded split-step methods optimized with a step size control for solving the femtosecond pulse propagation problem in the nonlinear fiber optics formalism

Abstract

We present the suitability of two optimized split-step methods for validating the femtosecond pulse propagation problem in the nonlinear fiber optics formalism that is governed by an extended nonlinear Schrödinger equation. In particular, the embedded split-step Fourier method (embedded-SSFM) and the embedded symmetrized split-step Fourier method (embedded-SymSSFM), which are optimized by the implementation of a step size control algorithm, are tested in terms of the femtosecond soliton fission phenomenology to analyze the numerical stability. As a result, it is possible to validate that these numerical methods introduce the appropriate resolution as function of the accuracy order that is needed at different stages of the soliton fission dynamics, which involves very large spectral shifts driven by the interaction between the nonlinear response and the high-order dispersion contribution of the optical fiber. Thus, the presented numerical methods can be used to validate accurately complex processes such as the development of supercontinuum spectra in the current formalism.

Quantum model for supercontinuum generation process

“…(8) is the same as the process for solving common GNLSE (classical treatment). Several numerical methods have been developed for solving the GNLSE equation 26 , 28 , 68 , 69 . For simulating both quantum-GNLSE and common GNLSE, the Fourier transform and inverse Fourier transform (i.e.…”

“…transformed the equation to the frequency domain then transformed back to the time domain) are applied in each integration step to treat the dispersion terms (the terms contain time derivatives). Also, the integral terms (convolution integrals) present in these equations can be evaluated accordingly by transforming each functions in the integrands into the frequency domain, multiplying, then transforming back 26 , 28 , 68 , 69 . Here, under the mean case condition, the main difference between two treatments is the additional term in Eq.…”

Comparative study of finite difference methods and pseudo-spectral methods for solving the nonlinear Schrödinger equation in optical fiber

“…In contrast, the numerical methods applied to the NLSE are far more suitable to reproduce a broad range of complex behaviors [5][6][7][8][9][11][12][13][14][15][16], as previously discussed. In general, both the finite difference methods and the pseudo-spectral methods are reported to integrate adequately different versions of nonlinear Schrödinger-type equations with a high degree of convergence and stability [2,5,[20][21][22][23][24][25][26]. Specifically, the finite difference methods are reported for solving nonlinear Schrödinger-type equations [20][21][22][23][24], which present similar mathematical expressions in comparison with the NLSE in fiber; however, the degree of convergence and stability of the numerical methods can be different between distinct fields of study that make use of the nonlinear Schrödinger-type equations, which involve different mathematical and physical pictures.…”

“…As a result, it is necessary that the analysis of the convergence and stability of the reported finite difference methods, which have not been studied exhaustively in the NLSE in fiber yet, be carried out in the study of the pulse propagation problem driven by the simultaneous action of the linear and nonlinear effects in optical fiber over long distances. On the other hand, the pseudo-spectral methods, which introduce two or more subdivisions at each step to integrate separately the linear and the nonlinear parts of the NLSE, have already been reported to validate the pulse propagation problem in fiber with an adequate degree of convergence and stability [25,26].…”

“…This feature of the finite difference methods is very attractive to solve nonlinear propagation equations where it can be difficult to separate the linear and nonlinear parts. Nonetheless, the NLSE in fiber can be readily split into its linear and nonlinear parts, so that there is not restriction to implement the pseudo-spectral methods in this case [25,26]. On the other hand, it is well-known that the finite difference methods, as implicit schemes, demand a high computational cost due to the large number of equations that are solved for each iterative step.…”