A short description of the ssfm method can be found on wikipedia. This method is ubiquitously used in engineering and physics applications. Pdf splitstep fourier method in modeling dual core nonlinear. These methods may reach second and third order of accuracy to provide an easier way to analyze the effects within the numerical approach. Plane wave stability of the splitstep fourier method for. Pdf an introduction to the split step fourier method using matlab. It is enabled by the fact that the period of oscillations of the unstable fourier modes is much smaller than the width of the soliton.
Parallel implementation of the splitstep fourier method for. The first, second and fourthorder versions of the method are presented. The widely used split step fourier method has difficulties when solving partial differential equations with saturable gain. Notes on the sine gordon equation david gablinger january 31, 2007 abstract in this seminar, we will introduce the sinegordon equation, and solve it using a baecklund transfomation. The symmetrized split step fourier method will be used for simulation in each coarse step, simulating the. The purpose of this chapter is to provide an introduction to a very powerful method in numerically solving the nlse, known as the splitstep fourier method. It is shown that using the sparsity property of the simulated signals. The proposed method depends on the idea of using a smaller number of spectral components compared to the classical splitstep fourier. Numerical methods for partial differential equations. The split step fourier method is developed and applied to migrating stacked seismic data in two and three dimensions. We analyze a numerical instability that occurs in the wellknown split step fourier method on the background of a soliton. This assumes that over a very small distance these components may be assumed to be independent. The split step fourier methodssfm is a part of the the family of. Su arez january 18, 20 abstract the split step fourier method provides an excellent methodology.
Stability analysis of the splitstep fourier method on the. We advance in time in small steps and treat separately the linear and nonlinear parts. Split step fourier method for solving the nlse equation 5 can now be solved using the ssfm. The linear subequation is still solved by the discrete fourier transform 17. We present a parallel version of the wellknown splitstep fourier method ssf for solving the nonlinear schrodinger equation, a mathematical model describing. The principle of the ssfm is simpler among them, it is easy to write program, and employed by many scholars. Computational fourier optics a matlab tutorial download. This method combines a fourier collocation in space with a strang splitting in time. Menyuk1 2 august 7, 2002 1department of computer science and electrical engineering, university of maryland baltimore county, baltimore, md 21250. The space variable is discretized by means of a finite difference and a. Analysis and characterization of fiber nonlinearities with.
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. A numerical method is therefore necessary for an understanding of the nonlinear effects in optical fibers. Click download or read online button to get computational fourier optics a matlab tutorial book now. First, we solve numerically the nonlinear subproblem.
Feb 11, 2009 hello all, i have short question at the end, but i wil gve short background. For example, it may be a chosen constant or a constant plus a small random perturbation that is uniformly distributed. In this talk, we present the simplicity of this method by solving the linear and nonlinear schr odinger equation and with. Sinkin et al optimization of the splitstep fourier method 63 where and is the number of steps per fiber span. Pdf splitstep fourier method in modeling dual core. An introduction to the split step fourier method using matlab pablo u. There are frequentlyused methods like perturbation method, differential method, split step fourier method ssfm et al. Splitstep method for generalized nonlinear equations. This migration method, which is implemented in both the frequencywavenumber. Robust splitstep fourier methods for simulating the propagation of ultrashort pulses in singleand twomode optical communication. Blue solid curves by split step fourier method and red dotted curves by variational method. It is shown that using the sparsity property of the simulated signals, the compressive sampling algorithm can be used as a very efficient tool for the splitstep spectral simulations. A numerical approach such as split step fourier and symmetrized split step fourier method were applied to the equation to have a better performance in terms of high orders accuracy. Abstract extensions of the splitstep fourier method ssfm for schrodingertype pulse propagation equations for simulating femtosecond pulses in single and.
In this paper we focus on the splitstep fourier method. A splitstep method is used to discretize the time variable for the numerical solution of the nonlinear schrodinger equation. New procedures are used to derive a scalar helmholtz equation and to subsequently transform to a rectangular coordinate system without requiring approximations. The split step fourier method ssfm is an efficient algorithm for the numerical solution of the nlse. Global simulation accuracy control in the splitstep fourier. The treatment, based on the technique of the split step fourier transform, enables us to avoid the uncontrolled approximations often used to solve the parabolic differential equation of the mutual coherence function. The step size with the coarse step method is designated in an.
Plane wave stability of the splitstep fourier method for the. Furthermore, we also give a numeric solution using a split step algorithm, and also present two physical applications of the sinegordon equation. 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 method is analyzed in split step methods for the solution of the nonlinear schrodinger equation by j. Robust splitstep fourier methods for simulating the propagation of. The present application of the split step fourier method to the gnls equation is set in a framework which differs in two aspects from that presented in 9. The finite difference method solves the maxwells wave equation explicitly in the timedomain under the assumption of the. Ultrafast soliton switching in a twocore fiber coupler is studied by controlling the coupling coefficients of the fiber. Pdf modified splitstep fourier method for the numerical. In the following paragraphs, we focus on split step fourier method as a calculation method and explain a method to analyze behaviours of pulse in an optical fiber. It is a special application of the splitting method for solving a pde. In 1d, an n element numpy array containing the intial values of \\psi\ at the spatial grid points.
Splitstep methods for the solution of the nonlinear. First, the method relies on computing the solution in small steps. In fact, noise from forwardpropagating amplified spontaneous emission, associated with a markov. In this paper, the ssfm is particularly introduced, and the friendly interface matlab software is. Pdf this is a tutorial on how to use a split step method using fourier transform with matlab. In 1822 he made the claim, seemingly preposterous at the time, that any function of t, continuous or discontinuous, could be represented as a linear combination of functions sinnt. We analyze a numerical instability that occurs in the well. An introduction to the split step fourier method using matlab. Split step fourier method file exchange matlab central. Sinkin,1 ronald holzlohner,1 john zweck,1 and curtis r.
The ssfm is the technique of choice for solving the nlse due to its easy implementation and speed compared to other methods, notably timedomain finitedifference methods 73. Generally, there are 2 ways to realize the ssfm, i. Splitstep fourier method in modeling dual core nonlinear fiber coupler james raja. This function performs the splitstep fourier method to solve the 1d timedependent schrodinger equation for a given potential. One dimensional problem of this equation has been explained in this. In this work we employ a split step fourier method to integrate the equation. The numerical analysis of soliton propagation with plitstep. This method is based around separating the dispersive and nonlinear components of the equation from one another. Using a modified version of the split step fourier method, we analyze the effect of noise on soliton propagation inside erbiumdoped fiber amplifiers.
For the love of physics walter lewin may 16, 2011 duration. Fourier splitting method for kawahara type equations. Compressive split step fourier method cihan bay nd r1, y 1department of civil engineering, i. Then the fourier series of ft is the series cc1a,1 cos iii h b7, sin itt, 18 i i where the fourier. A reliable split step fourier method for the propagation equation of ultrafast pulses in singlemode optical fibers ralf deiterding, roland glowinski, hilde oliver, and stephen poole member, ieee abstractthe extension to the split step fourier method ssfm for. Plane wave stability of the splitstep fourier method 3 on long time intervals of the exact solution near plane waves 8 is shared by one of the most popular numerical methods for the nonlinear schrodinger.
Some of the numerical methods are euler method, modified euler method, higher order rungekutta method, finite difference method, split step fourier transform method. Optical soliton simulation by symmetrized splitstep. Fourier series jean baptiste joseph fourier 17681830 was a french mathematician, physicist and engineer, and the founder of fourier analysis. 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. We present a parallel version of the wellknown splitstep fourier method ssf for solving the nonlinear schrodinger equation, a mathematical model. By employing the splitstep fourier method ssfm, dbp can solve the nonlinear schrodinger equation nlse inversely and hence the optical signals can be reconstructed at the receiver side.
The step split fourier method ssfm provides an excellent methodology for learning and teaching how to solve time dependent partial di erential equations. 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. Split step fourier method, numerical instability, nonlinear evolution equations, solitary. Split step fourier propagation beam propagation method. Pdf compressive splitstep fourier method researchgate. Our main result states that the longtime orbital stability of the exact solution. Parallel implementations of the splitstep fourier method for solving.
Robust splitstep fourier methods for simulating the. Convergence of splitstep generalizedlaguerrefourier. Walkoff method in many opticalfiber communications systems, chromatic dispersion is the dominant effect and nonlinearity only plays a. Apr 16, 2012 split step fourier method set 2 jeff wong. Contribute to eeewxjmatlab development by creating an account on github. The subject is regarding the split step fourier method ssfm adn i will be gratefull if someone who know the method can help. Comparison of numerical methods for modeling laser mode. Pdf parallel implementations of the splitstep fourier method for. The proposed method depends on the idea of using a smaller number of spectral components compared to the classical splitstep fourier method with a high. Pdf an introduction to the split step fourier method. On a faster parallel implementation of the splitstep fourier. 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. Laserfoam laser pulse propagation using a fourier adaptive. Generally speaking, the method is useful to face out the problem 43.
The chapter will begin with a list pointing the advantages of the ssfm. Splitstep fourier method how is splitstep fourier method. Optical soliton simulation by symmetrized splitstep fourier. An integral splitstep fourier method for digital back. A theoretical foundation for the use of the parabolic wave equation fourier split. Convergence of split step generalizedlaguerre fourier hermite methods for grosspitaevskii equations with rotation term harald hofst atter othmar koch mechthild thalhammer august 3, 2012 abstract a convergence analysis for timesplitting generalizedlaguerre fourier hermite pseudospectral methods applied to timedependent gross. The splitting algorithm is a pseudospectral numerical method which consists in approximate iteratively the solution of the nonlinear evolution equation using exponential operators. Parallel splitstep fourier methods for the cmkdv equation t. This migration method, which is implemented in both the frequencywavenumber and frequencyspace domains, takes into account laterally varying velocity by defining a reference slowness reciprocal of velocity as the mean slowness in the migration interval and a perturbation. The numerical investigation of all optical soliton switching is done by using split step fourier transformation algorithm. Apr 16, 2012 for the love of physics walter lewin may 16, 2011 duration. Apr 04, 2016 i am trying to write a very basic matlab code to preform the split step fourier method on the nonlinear schrodinger equation. The splitstep fourier ssf method, commonly used in the numerical solution of the nlse, often proves too slow in serial versions, even on the fastest.
We will call this implementation of the split step method the logarithmic step size method. Our main result states that the longtime orbital stability of the exact solution near plane waves transfers to the numerical solution, see subsect. Compressive splitstep fourier method cihan bay nd r1, y 1department of civil engineering, i. This instability is found to be very sensitive to small. The splitstep fourier method i am asked to use works by writing 1 as. This method combines a fourier collocation in space with a strang splitting in time, see subsect. Fft beam propagation method file exchange matlab central. A variation on this method is the symmetrized split step fourier method, which takes half a time step using one operator, then takes a fulltime step with only the other, and then takes a second half time step again with only the first. Sep 17, 20 the zip file includes the next programs.
View enhanced pdf access article on wiley online library html view download pdf for offline viewing. Poole abstract extensions of the splitstep fourier method ssfm for schrodingertype. This site is like a library, use search box in the widget to get ebook that you want. Figure 33 normalized output widths as a function of normalized input width so at three distances. A new parallel implementation of the split step fourier method is presented. Theoretical description of the parabolic approximation. Proposed method can be named as compressive splitstep fourier method cssfm. A new method of obtaining the mutual coherence function of a laser pulse propagating in a particulate medium is described. For testing of the proposed method the nonlinear schrodinger equation and its onesoliton and twosoliton solutions. Splitstep fourier migration geophysics geoscienceworld.
Here, we describe a modified split step fourier method, and we compare it to several different algorithms for solving the haus modelocking equation and related equations that are used to model modelocked. Optimization of the splitstep fourier method in modeling optical fiber communications systems oleg v. Fourier transform, fourier series, and frequency spectrum duration. Timedependent schrodinger equation via the splitstep. In this paper an approach for decreasing the computational effort required for the splitstep fourier method ssfm is introduced. The computational core of our scheme focuses on the parallelization of the whole linear step using a combination of fast cyclic convolutions and fast fourier transforms. Pdf an introduction to the split step fourier method using. Optimization of the splitstep fourier method in modeling. Sorry that this post is long but i am trying to simulate two dimensional schrodinger equation in python using split step method.
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