Split step fourier method pdf

First of all, we deform the nonlinear schroedinger equation into the following expression. Splitstep fourier method in modeling dual core nonlinear fiber coupler james raja. Parallel numerical methods for solving nonlinear evolution. Compressive split step fourier method cihan bay nd r1, y 1department of civil engineering, i. In this talk we employ the well known split step fourier method for the numerical simulation of the nls and cnls equations. We present a parallel version of the wellknown splitstep fourier method ssf for solving the nonlinear schrodinger equation, a mathematical model. We analyze a numerical instability that occurs in the wellknown split step fourier method on the background of a soliton. Splitstep fourier method for nonlinear schrodinger. Split step fourier method file exchange matlab central. The numerical singularity induced by the reflection coefficient at pseudobrewster incidence is analyzed, and the dmft solution for ssflbc resolution is. The efficiency of the split step method depends on both the timedo. Timedependent schrodinger equation via the splitstep. Jun 02, 2006 splitstep fourier method for nonlinear schrodinger equation. 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 finite.

Splitstep fourier method for nonlinear schrodinger equation. 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. Then the fourier series of ft is the series cc1a,1 cos iii h b7, sin itt, 18 i i where the fourier coefficients a, and b, are defined by means of the. Split step fourier propagation beam propagation method. Sorry that this post is long but i am trying to simulate two dimensional schrodinger equation in python using splitstep method. In fact, noise from forwardpropagating amplified spontaneous emission, associated with a markov. There are frequentlyused methods like perturbation method, differential method, splitstep fourier method ssfm et al. Proposed method can be named as compressive splitstep fourier method cssfm. 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. The representation of nlse using splitstep fourier method ssfm 3. Forward propagation above dielectric surfaces is studied using the split step fourier ssf resolution technique. Parallel splitstep fourier methods for the cmkdv equation t. Pdf stability analysis of the splitstep fourier method on.

Abstract extensions of the splitstep fourier method ssfm for schrodingertype pulse propagation equations for simulating femtosecond pulses in single and. Plane wave stability of the splitstep fourier method for the. The numerical investigation of all optical soliton switching is done by using split step fourier transformation algorithm. Assuming i want to propagate a gaussian in free space, im supposed to use. We analyze a numerical instability that occurs in the wellknown splitstep fourier method on the background of a soliton. Sinkin et al optimization of the split step fourier method 63 where and is the number of steps per fiber span. 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. Various versions of the splitstep fourier method ssfm for the nonlinear.

Fourier splitting method for kawahara type equations. Compressive splitstep fourier method cihan bay nd r1, y 1department of civil engineering, i. Pdf splitstep fourier method in modeling dual core. Using a modified version of the split step fourier method, we analyze the effect of noise on soliton propagation inside erbiumdoped fiber amplifiers. In this paper, we focus on the split step fourier method. Pdf an introduction to the split step fourier method using. An introduction to the split step fourier method using matlab. We also present a parallelization of the split step fourier method using the fastest fourier. Plane wave stability of the splitstep fourier method for. We will discuss the fastfouriertransform method, which should be used to e ciently carry out the long series of fourier and inverse fourier transformations needed to propagate the wave function this way for a large number of time steps. In this work we employ a splitstep fourier method to integrate the equation. Using a modified version of the splitstep fourier method, we analyze the effect of noise on soliton propagation inside erbiumdoped fiber amplifiers.

Propagation modeling using the split step fourier method. We analyze a numerical instability that occurs in the well. Theoretical description of the parabolic approximation. 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 split step spectral simulations of various phenomena which can be modeled by using differential equations. We will call this implementation of the splitstep method the logarithmic stepsize method. The principle of the ssfm is simpler among them, it is easy to write program, and employed by many scholars. In order to explain the split step fourier method, we use the nonlinear schroedinger equation.

Download pdf download citation view references email request permissions export to collabratec. Hanik, fast implementation of the split step fourier method using a graphics processing unit, in optical fiber communication conference, osa technical digest cd optical society of america, 2010, paper otud7. 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 of various phenomena which can be modeled by using differential equations. For testing of the proposed method the nonlinear schrodinger equation and its onesoliton and twosoliton solutions. In this paper we focus on the split step fourier method. 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. Study of instability of the fourier splitstep method for the. The numerical analysis of soliton propagation with plit.

Our main result states that the longtime orbital stability of the exact solution near plane waves transfers to the numerical solution, see subsect. This method combines a fourier collocation in space with a strang splitting in time. 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 reliable splitstep 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 splitstep fourier method ssfm. Plane wave stability of the splitstep fourier method for the nonlinear schr odinger equation erwan faou1,2 ludwig gauckler3 christian lubich4 version of 2 dezember 20 abstract plane wave solutions to the cubic nonlinear schr odinger equation on a torus.

Osa fast implementation of the splitstep fourier method. Ultrasound imaging of long bone fractures and healing with. We compared the performance of the different implementations for a variety of pulse formats and systems,including higher order solitons, collisions of soliton pulses, a singlechannel periodically. In this paper an approach for decreasing the computational effort required for the split step fourier method ssfm is introduced. Stability properties of the wellknown fourier splitstep method used to simulate a soliton and similar solutions of the nonlinear dirac equations, known as the grossneveu model, are studied numerically and analytically. One dimensional problem of this equation has been explained in this post. Pdf an introduction to the split step fourier method using matlab. In this method, the equation is discretized in space by a spectral collocation method and in time by a splitting integrator. 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. Numerical solutions of the schr odinger equation 1. The splitstep fourier method is used in threedimensional parabolicequation pe models to compute underwater sound propagation in one direction i. Three distinct types of numerical instability that can occur in this case, are revealed and explained. Splitstep methods for the solution of the nonlinear. Pdf this is a tutorial on how to use a split step method using fourier transform with matlab.

We set up different long bone models for numerical simulation. The linear subequation is still solved by the discrete fourier transform 17. Generalizations i bregman iteration can be used to solve a wide range of nonl1 problems argminju such that au 0 where j and kak2 are convex. This function performs the splitstep fourier method to solve the 1d timedependent schrodinger equation for a given potential. A theoretical foundation for the use of the parabolic wave equationfourier split. Furthermore, we also give a numeric solution using a split step algorithm, and also present two physical applications of the sinegordon equation. A higherorder splitstep fourier parabolicequation sound.

Proposed method can be named as compressive split step fourier method cssfm. Parallel implementations of the splitstep fourier method for solving. In this work, we used the splitstep fourier imaging method to reconstruct bone models. Hanik, fast implementation of the splitstep fourier method using a graphics processing unit, in optical fiber communication conference, osa technical digest cd optical society of america, 2010, paper otud7. The chapter will begin with a list pointing the advantages of the. Pdf in this paper an approach for decreasing the computational effort required for the splitstep fourier method ssfm is introduced. Parabolic equation method in underwater acoustics, j. Stability properties of the wellknown fourier split step method used to simulate a soliton and similar solutions of the nonlinear dirac equations, known as the grossneveu model, are studied numerically and analytically.

Sorry that this post is long but i am trying to simulate two dimensional schrodinger equation in python using split step method. Sinkin et al optimization of the splitstep fourier method 63 where and is the number of steps per fiber span. In this talk we employ the well known splitstep fourier method for the numerical simulation of the nls and cnls equations. Fast fourier transform history twiddle factor ffts noncoprime sublengths 1805 gauss predates even fouriers work on transforms. Pdf an introduction to the split step fourier method. The finite difference method solves the maxwells wave equation explicitly in the timedomain under the assumption of the. To apply the new method, we considered a complicated system of 16channel wavelengths, where the splitstep fourier method was used to solve the nonlinear schrodinger equations, and the transfer matrix method was applied to describe the coupled mode equations. Numerical methods for partial differential equations. The method is implemented in both cartesian x, y, z and cylindrical r. Optimization of the splitstep fourier method in modeling. Then the fourier series of ft is the series cc1a,1 cos iii h b7, sin itt, 18 i i where the fourier. Furthermore, we also give a numeric solution using a splitstep algorithm, and also present two physical applications of the sinegordon equation.

This method combines a fourier collocation in space with a strang splitting in time, see subsect. Splitstep method for generalized nonlinear equations. The method is widely used in geophysics to image the earth interior, but to our knowledge has not traditionally been used for bone imaging. Numerical solutions of the schr odinger equation 1 introduction. Pdf parallel implementations of the splitstep fourier method for.

Pdf modified splitstep fourier method for the numerical. I am trying to write a very basic matlab code to preform the splitstep fourier method on the nonlinear schrodinger equation. The introduction of fresnel boundary conditions ssffbc and leontovitch boundary conditions ssflbc is described. Download pdf download citation view references email request permissions export to collabratec alerts metadata. A splitstep method is used to discretize the time variable for the numerical solution of the nonlinear schrodinger equation. In ffd method, usually the minimum velocity is used as the reference velocity.

Hello im having a difficulty using the split step fourier method. Oct 27, 2010 we analyze a numerical instability that occurs in the well. Robust splitstep fourier methods for simulating the. Split step method in the analysis and modeling of optical fiber communication system. Study of instability of the fourier splitstep method for. Parallel splitstep fourier methods for the cmkdv equation. 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. We advance in time in small steps and treat separately the linear and nonlinear parts. One of the most popular numerical methods to solve the perturbed nlse is the splitstep fourier method ssf 2. Abstract in this paper an approach for decreasing the computational effort required for the split step fourier method ssfm is introduced. We will discuss the fast fourier transform method, which should be used to e ciently carry out the long series of fourier and inverse fourier transformations needed to propagate the wave function this way for a large number of time steps. In numerical analysis, the splitstep fourier method is a pseudospectral numerical method used to solve nonlinear partial differential equations like the.

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. A short description of the ssfm method can be found on wikipedia. We will call this implementation of the split step method the logarithmic step size method. Pdf stability analysis of the splitstep fourier method.

Walkoff method in many opticalfiber communications systems, chromatic dispersion is the dominant effect and nonlinearity only plays a. The splitting algorithm is a pseudospectral numerical method which consists in approximate iteratively the solution of the nonlinear evolution equation using exponential operators. In this paper an approach for decreasing the computational effort required for the splitstep fourier method ssfm is introduced. Theoretical analysis of a novel wdm optical longhaul. Pdf splitstep fourier method in modeling dual core nonlinear. Theoretical analysis of a novel wdm optical longhaul network. In 1d, an n element numpy array containing the intial values of \\psi\ at the spatial grid points.

The space variable is discretized by means of a finite difference and a. The split bregman method for l1regularized problems. Ultrafast soliton switching in a twocore fiber coupler is studied by controlling the coupling coefficients of the fiber. 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. Numerical method and statement of the main results 2. 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. Osa optimization of the splitstep fourier method in. In this paper, the ssfm is particularly introduced, and the friendly interface matlab software is. In this paper we focus on the splitstep fourier method.

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