We will call this implementation of the splitstep method the logarithmic stepsize method. A algorithm used for simulate a pulse propagation in fiber optics jeovazerosplitstepfouriermethodpython. Robust splitstep fourier methods for simulating the propagation of. Menyuk1 2 august 7, 2002 1department of computer science and electrical engineering, university of maryland baltimore county, baltimore, md 21250. First, the method relies on computing the solution in small steps, and treating the linear and the nonlinear steps separately see below. Limitation of splitstep fourier method physics forums. This instability is found to be very sensitive to small changes of the parameters of both the numerical grid and the soliton, unlike the instability of most finitedifference schemes. The application was made using the python programming language, and the libraries numpy, scipy, pygtk and matplotlib.
An introduction to the split step fourier method using matlab. Using a splitstep fourier method for a radial equation. Splitstep method for generalized nonlinear equations. Sinkin et al optimization of the splitstep fourier method 63 where and is the number of steps per fiber span. The precision of numerical simulation by using assfm is higher than that by using splitstep fourier method in the process of laser pulse propagation in pcfs due to the fact that the variation of fibre parameters with the peak. Performancewise, gpuwave has been found to be up to three orders of magnitude faster than peec, a comparable cpubased implementation of the splitstep fourier method, and the. The linear subequation is still solved by the discrete fourier transform 17. Split step method for nonlinear schrodinger equation does not result in self focusing. A algorithm used for simulate a pulse propagation in fiber optics jeovazerosplitstep fourier method python. Splitstep quasispectral finite difference method for nonlinear optical pulse propagation 2006 optical fiber communication conference and the national fiber optic engineers conference published. Sorry that this post is long but i am trying to simulate two dimensional schrodinger equation in python using splitstep method. It makes use of a modal method to solve the electromagnetic problem. Parallel splitstep fourier methods for the cmkdv equation.
Robust splitstep fourier methods for simulating the propagation of ultrashort pulses in singleand twomode optical communication. Is there some conditions under which the split step method breaks down. In order to solve the nonlinear schodinger equation numerically, the split step fourier method can be used. This function performs the splitstep fourier method to solve the 1d timedependent schrodinger equation for a given potential. In this study, the complex modified kortewegde vries cmkdv equation is solved numerically by three different splitstep fourier schemes. Animating the schrodinger equation wed 05 september 2012.
Ultrafast soliton switching in a twocore fiber coupler is studied by controlling the coupling coefficients of the fiber. An algorithm used for simulate a pulse propagation in fiber optics. Splitstep fourier method for nonlinear schrodinger. In numerical analysis, the splitstep fourier method is a pseudospectral numerical method used to solve nonlinear partial differential equations like the. Note that the actual propagation of the pulse takes up just a. Stability analysis of the splitstep fourier method on the. Split step fourier method for solving the nlse equation 5 can now be solved using the ssfm. A comparison of the mapped fouriersine method with. We analyze a numerical instability that occurs in the wellknown splitstep fourier method on the background of a soliton.
Fourier transform, fourier series, and frequency spectrum duration. Adaptive splitstep fourier method for simulating ultrashort laser pulse propagation in photonic crystal fibres. An algorithm used for simulate a pulse propagation in fiber optics jeovazerosplitstepfouriermethodpython. I need to show that a optical pulseform of a hyperbolic secant is a. First of all, we deform the nonlinear schroedinger equation into the following expression. In numerical analysis, the splitstep fourier method is a pseudospectral numerical method used to solve nonlinear partial differential equations like the nonlinear schrodinger equation. Various versions of the splitstep fourier method ssfm for the nonlinear schrodinger equation nlse are presented. Fft beam propagation method file exchange matlab central. But before implement this method in matlab, i would like to ask if this method always works in. In 1d, an n element numpy array containing the intial values of \\psi\ at the spatial grid points. This method combines a fourier collocation in space with a strang splitting in time.
Description of dispersive wave emission and supercontinuum generation in silicon waveguides using splitstep fourier and rungekutta integration methods. The split step fourier method is an alternative to the. The integration is performed using a firstorder splitstep pseudospectral method, relying on the fast fourier transform. Walkoff method in many opticalfiber communications systems, chromatic dispersion is the dominant effect and nonlinearity only plays a. A detailed description of equations used in this program, as well as the splitstep fourier method ssfm can be found on the book nonlinear fiber optics by g. Optimization of the splitstep fourier method in modeling. This software running on windows and macos comes with a gui that permits to define with the aid of files or scripts the arbitrary and complex geometry of. Mathematica code for the split step fourier method. An introduction to the split step fourier method using matlab free download as pdf file. Split step fourier method file exchange matlab central.
View academics in splitstep fourier method on academia. This method occurs in several applications, and is a useful numerical method when the equation can be split into two separate equations, each of which can either be solved exactly, or each part is best solved by a different numerical method. The splitting introduces high frequent oscillations in the intermediate steps which do not disappear after a full time step is completed. The numerical investigation of all optical soliton switching is done by using split step fourier transformation algorithm. We perform a systematic study of the accuracy of splitstep fourier transform methods for the time dependent grosspitaevskii equation using symbolic. Split step method for nonlinear schrodinger equation does. One dimensional problem of this equation has been explained in this post. Timedependent schrodinger equation via the splitstep. In this method, the equation is discretized in space by a spectral collocation method and in time by a splitting integrator. The solution to equation 6 that is used by the splitstep fourier method is based on ignoring the ul contribution and can be summarized as follows. Petrowiki was initially created from the seven volume petroleum engineering handbook peh published by the society of petroleum engineers spe the seg wiki is a useful collection of information for working geophysicists, educators, and students in the field of geophysics. A detailed description of equations used in this program, as well as the splitstep fourier method ssfm can be found. It provides a graphical environment to run and visualize the results.
I am trying to write a very basic matlab code to preform the splitstep fourier method on the nonlinear schrodinger equation. We will consider a numerical method to solve this equation known as splitting. Well approach this using the splitstep fourier method. Split step fourier propagation beam propagation method. A splitstep fourier method for the complex modified. An open source virtual laboratory for the schrodinger. I am going to use this method to solve a nonlinear schrodinger equation with variable coefficients.
Because the split step fourier method employs an algorithm of fast fourier transform fft, the analysis speed is faster than most of other analytical methods. You can download it at the sourceforge project page. The approach for comparison is to a fix the accuracy. Parallel spectral numerical methodsthe cubic nonlinear. Also, it features methods for simulating both threewavemixing processes such as dfg. The main difference among the three schemes is in the order of the splitting approximation used to factorize the exponential operator. Laserfoam laser pulse propagation using a fourier adaptive. Fourier transform the previously migrated upgoing wave field at depth zn pr, z.
Pdf splitstep fourier method in modeling dual core. Proposed method can be named as compressive splitstep fourier method cssfm. If i consider only nonlinear effect in this code, the shape of pusle in time domain does not change righttheory says that nonlinear effects has direct influence on pulse in spectral domain spectral broadening, but inderect influence on pulse shape in time domain. Python implementation of the splitstep fourier spectral method to solve the grosspitaevski equation shreyaspotnisgpe1d. The space variable is discretized by means of a finite difference and a fourier method. Robust splitstep fourier methods for simulating the. The chapter will begin with a list pointing the advantages of the ssfm. The singular coulomb potential introduces additional difficulties when split step methods are used.
The application was made using the python programming language, and the libraries numpy. Rsm radiation spectrum method is a 2d rigorous tool to solve the maxwell equations for the propagation of light in integrated optics or photonics devices. Migration using the splitstep fourier method onepetro. Extensions of the splitstep fourier method ssfm for schrodingertype pulse propagation equations for simulating femtosecond. This method is based around separating the dispersive and nonlinear components of the equation from one another. Siam journal on numerical analysis siam society for. In the case of python, the suggestion is to download the latest. Create scripts with code, output, and formatted text in a single executable document. In order to explain the split step fourier method, we use the nonlinear schroedinger equation. Hi, im using the splitstep fourier method for programming.
This assumes that over a very small distance these components may be assumed to be independent. Plane wave stability of the splitstep fourier method for. A simple pythonbased open source software library for the numerical. Alternatively, you can download the latest version from the pynlo github site.
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