Study on soliton in photorefractive crystals has involved a variety of disc. Because it is often more efficient than finite differences for the simulation of nonlinear schrodinger nls systems 11, the ssf method is the more commonly used. Splitstep fourier migration geophysics geoscienceworld. Ssf is the most efficient technique because it uses fast. Using local born and local rytov fourier modeling and. The widely used split step fourier method has difficulties when solving partial differential equations with saturable gain. The splitstep fourier method is a powerful technique for solving partial differential equations and simulating ultracold atomic systems of various. Plane wave stability of the splitstep fourier method for. Proposed method can be named as compressive split step fourier method cssfm. Optical soliton simulation by symmetrized splitstep. Sinkin et al optimization of the split step fourier method 63 where and is the number of steps per fiber span. Ultrasound pulseecho imaging using the splitstep fourier. 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.
A wideangle split step algorithm for the parabolic equation, j. The finite difference method solves the maxwells wave equation explicitly in the timedomain under the assumption of the. Application of the splitstep fourier method in investigating. These methods are analyzed with respect to various physical properties. When there is a balance between gvd and spm, there exists a stable wave optical soliton which can propagate for long distance undistorted. R assessment of higherorder exponential operators for the simulation of high capacity optical communication systems by the split step fourier method. The efficiency of the split step method depends on both the timedo. Mathematically equation 6 is given by, 8 the accuracy of the split step fourier method can be improved by adopting a different procedure to propagate optical pulse over one segment from to. 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. In numerical analysis, the splitstep fourier method is a pseudospectral numerical method used to solve nonlinear partial differential equations like the. Pdf an introduction to the split step fourier method using matlab. 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.
Split step method in the analysis and modeling of optical fiber. Fourier series jean baptiste joseph fourier 17681830 was a french mathematician, physicist and engineer, and the founder of fourier analysis. Let us then generalize the fourier series to complex functions. So the numerical methods are introduced to simulate the propagation of the solitons. Accuracy analysis of the splitstep fourier propagator. New procedures are used to derive a scalar helmholtz equation and to subsequently transform to a rectangular coordinate system without requiring approximations. Submitted to journal of lightwave technology 1 a reliable. Theoretical description of the parabolic approximation. This method can simultaneously achieve a higher simulation accuracy and a slightly reduced complexity. Abstract we analyze a numerical instability that occurs in the well.
Parallel splitstep fourier methods for the coupled nonlinear. We advance in time in small steps and treat separately the linear and nonlinear parts. We investigate a modified split step fourier method ssfm by including lowpass filters in the linear steps. Our numerical experiments have shown that these methods give accurate results and considerable speedup. Three distinct types of numerical instability that can occur in this case, are revealed and explained. A splitstep fourier method for the complex modified kortewegde. Gpuimplementation of the split step fourier method with cuda within the scope of this thesis, the split step fourier method ssfm, a numerical method solving the nonlinear schrodinger equation nlse, will be implemented. A classical problem concerning the motion of a single solitary wave is used to compare the. Propagation modeling using split step fourier method core. The split step fourier method is a powerful technique for solving partial di. Hello im having a difficulty using the split step fourier method. The split step fourier propagator is a oneway wave propagation method that has been widely used to simulate primary forward and backward reflected deterministicrandom wave propagation due to its fast computational speed and limited computer memory requirement.
A higherorder splitstep fourier parabolicequation sound. The authors have been investigating a suite of modeling and migration methods that are implemented in the wavenumberspace domains and operate on data in the frequency domain. Pdf optimization of the splitstep fourier method in. Pdf acceleration of the splitstep fourier method by using. The split step fourier method is developed and applied to migrating stacked seismic data in two and three dimensions. A short description of the ssfm method can be found on wikipedia. Plane wave stability of the splitstep fourier method for the.
Parallel splitstep fourier methods for the cmkdv equation t. Pdf compressive splitstep fourier method researchgate. Modeling the longrange wave propagation by a splitstep. Pdf in this paper an approach for decreasing the computational effort required for the split step fourier method ssfm is introduced. Apr 18, 2008 for the trapezoidalbackward difference split step method, the text refers on page 179 to this graph of stability regions from the paper optimal stability for trapezoidalbackward difference split steps by sohan dharmaraja, yinghui wang, and gilbert strang, ima journal of numerical analysis 2009. Fourier transforms and the fast fourier transform fft algorithm paul heckbert feb. Siam news, volume 32, number 1 parallel implementation of the. These methods may reach second and third order of accuracy to provide an easier way to analyze the effects within the numerical approach.
In this study, the optical soliton simulation will be carried out using symmetrized splitstep fourier method ssfm. The implementation has to be conducted, using the cuda framework for parallel implementation on graphical processing units. The principle of the ssfm is simpler among them, it is easy to write program, and employed by many scholars. Fourier transforms and the fast fourier transform fft. This paper investigates the impact on the optical signaltonoise ratio osnr of the residual per span rdps in a n. The numerical analysis of soliton propagation with plitstep. Among those algorithms, the split step fourier method tappert, 1974a 8. To motivate this, return to the fourier series, eq. Split step fourier propagation beam propagation method. Study of instability of the fourier splitstep method for the. Pdf acceleration of the splitstep fourier method by. Fourier transforms and the fast fourier transform fft algorithm.
Feb 01, 2020 a split step wavelet method for simulating the longrange wave propagation is introduced. One major advantage of the split step fourier method is that it is based on the fourier transform, and therefore, the numerical dispersion is minimized. The splitstep fourier method is used in threedimensional parabolicequation pe models to compute underwater sound propagation in one direction i. Optical soliton simulation by symmetrized splitstep fourier. The splitting algorithm is a pseudospectral numerical method which consists in approximate iteratively the solution of the nonlinear evolution equation using exponential operators. 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. In this paper we introduce parallel split step fourier methods for the numerical simulations of the coupled nonlinear schrodinger equation that. Compared to other nlse models, ssfm is two magnitude. Feb 01, 2020 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. An open source virtual laboratory for the schrodinger equation. Pdf splitstep fourier method in modeling dual core. The method is implemented in both cartesian x, y, z and cylindrical r. In section 3, we introduce the parallel algorithm for ssf.
In addition a comparison among the algorithms is discussed. When using an assumption of uniform soundspeed for image reconstruction v v0, the method leads to the phaseshift image reconstruction scheme. Assuming i want to propagate a gaussian in free space, im supposed to use. Oct 27, 2010 we analyze a numerical instability that occurs in the well. 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.
The split step fourier method is a well know and powerful method in numerically solving the nonlinear schrodinger equation 101, especially for pulsepropagation problems in nonlinear dispersive. Pdf evaluation of nonlinear interference effects in a. Fig 3c shows the image using complex pade migration method for the exploding reflector data set. We look at a spike, a step function, and a rampand smoother functions too.
Robust splitstep fourier methods for simulating the. What i want to do is plot ix,z0, ix,z3 and the intensity distribution in the xz plane. Split step fourier method the ssf method is used to integrate many types of nonlinear partial differential equations. In pwe resolution, classically, boundary conditions at the ground limit is introduced via the leontovitch impedance boundary condition 9. Using a modified version of the splitstep fourier method, we analyze the effect of noise on soliton propagation inside erbiumdoped fiber amplifiers.
In this work we employ a split step fourier method to integrate the equation. Fourier splitting method for kawahara type equations. Splitstep methods for the solution of the nonlinear. 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. Robust splitstep fourier methods for simulating the propagation of. Improved simulation accuracy of the splitstep fourier method. An introduction to the split step fourier method using matlab free download as pdf file.
Modified splitstep fourier method for the numerical simulation of. In order to solve the nonlinear scho dinger equation numerically, the split step fourier method can be used. Split step fourier methods in this section we presentthreesplitstepfourierschemesforthe cmkdv equation 1. Pdf an introduction to the split step fourier method. Powerful propagationprediction tools, based on the split step fourier transform and the finiteelement method fem solutions of the parabolic equation pe are discussed. Numerical solutions of the kortewegde vries equation and its generalizations by the split step fourier method, in nonlinear wave motion, edited by a. Walkoff method in many opticalfiber communications systems, chromatic dispersion is the dominant effect and nonlinearity only plays a. In this body of work, we focus on several variations of this method to allow for simulations of one.
Thanks very much, i am a student major in solve nls with splitstep fourier method. Parallel implementations of the splitstep fourier method. Gpuimplementation of the splitstep fourier method with cuda. We analyze a numerical instability that occurs in the wellknown split step fourier method on the background of a soliton. First, the method relies on computing the solution in small steps, and treating the linear and the nonlinear steps separately see below. Jul 14, 2006 a split step method is used to discretize the time variable for the numerical solution of the nonlinear schrodinger equation. Pdf this is a tutorial on how to use a split step method using fourier transform with matlab. The best known example of these methods is the split step fourier method ssf. Originally developed for studyi ng the propagation of intense laser beams through.
Propagation modeling using split step fourier method. 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. To solve pwe, the split step fourier method ssf is the most efficient technique because it uses fast fourier transform fft and permits large step in range. A theoretical foundation for the use of the parabolic wave equation fourier split. Comparison of numerical methods for modeling laser mode. Massively parallel splitstep fourier techniques for. Splitstep complex padefourier depth migration geophysical.
We perform a systematic study of the accuracy of split step fourier transform methods for the time dependent grosspitaevskii equation using symbolic. This approximation, denoted by q 2, can also work with the split step fourier method, and it was first used by thomson and chapman 1983 12. The split step fourier method is a powerful technique for solving partial differential equations and simulating ultracold atomic systems of various. Suppression of reflection from the grid boundary in solving the time. 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. Parallel implementations of the splitstep fourier method for. Solitons in photorefractive crystals have been known to exist and currently become a trend of research in the modern era. Pdf an introduction to the split step fourier method using. In this method, the equation is discretized in space by a spectral collocation method and in time by a splitting integrator. The parabolic equation represents oneway propagation, and is widely used in twodimensional 20 groundwave propagation modeling. Splitstep fourier methods for the grosspitaevskii equation. The space variable is discretized by means of a finite difference and a fourier method. Siam news, volume 32, number 1 parallel implementation of.
The linear part is solved using the fourier transform and the nonlinear part is solved with the aid of the exponential operator. 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. These methods are analyzed with respect to various physical properties represented in the nls. Splitstep methods for the solution of the nonlinear schrodinger. 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 computational time of the split step complex pade fourier migration increases 10 per cent compared to the ffd method and the memory usage is the same for both methods. Split step fourier method file exchange matlab central.
Split step method in the analysis and modeling of optical. Timing results and conclusions are given in sections 4 and 5, respectively. 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. In this paper, we focus on the split step fourier method. Fourier splitting method for kawahara type equations hindawi. There are frequentlyused methods like perturbation method, differential method, split step fourier method ssfm et al. We will call this implementation of the split step method the logarithmic step size method. An introduction to the split step fourier method using matlab.
In 1822 he made the claim, seemingly preposterous at the time, that any function of t, continuous or discontinuous, could be. Parallel splitstep fourier methods for the cmkdv equation. Stability analysis of the splitstep fourier method on the. Optimization of the splitstep fourier method in modeling. Extensions of the split step fourier method ssfm for schrodingertype pulse propagation equations for simulating. Nonlinear schrodinger equation split step fourier transform dispersion fiber nonlinearities chirp. Compared to the split step fourier method, this method improves the computation efficiency while keeping a good accuracy. 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 schr. In this study, the complex modified kortewegde vries cmkdv equation is solved numerically by three different split step fourier schemes. In this paper we focus on the split step fourier method. In this method, one complete marching step of the solution is split into two sub steps to sequentially, not simultaneously, solve for free space propagation and.
1271 1401 167 456 1453 1341 1080 463 341 564 1499 698 1116 16 694 1273 428 358 1112 65 1132 342 1466 438 824 425 1479 853 504 715