Split-step method

In numerical analysis, the split-step (Fourier) method is a pseudo-spectral numerical method used to solve nonlinear partial differential equations like the nonlinear Schrödinger equation. The name arises for two reasons. First, the method relies on computing the solution in small steps, and treating the linear and the nonlinear steps separately (see below). Second, it is necessary to Fourier transform back and forth because the linear step is made in the frequency domain while the nonlinear step is made in the time domain.

An example of usage of this method is in the field of light pulse propagation in optical fibers, where the interaction of linear and nonlinear mechanisms makes it difficult to find general analytical solutions. However, the split-step method provides a numerical solution to the problem.

Description of the method

Consider, for example, the nonlinear Schrödinger equation

${\partial A \over \partial z}=-{i\beta _{2} \over 2}{\partial ^{2}A \over \partial t^{2}}+i\gamma |A|^{2}A=[{\hat {D}}+{\hat {N}}]A,$

where $A(t,z)$ describes the pulse envelope in time $t$ at the spatial position $z$. The equation can be split into a linear part,

${\partial A_{D} \over \partial z}=-{i\beta _{2} \over 2}{\partial ^{2}A \over \partial t^{2}}={\hat {D}}A,$

and a nonlinear part,

${\partial A_{N} \over \partial z}=i\gamma |A|^{2}A={\hat {N}}A.$

Both the linear and the nonlinear parts have analytical solutions, but the nonlinear Schrödinger equation containing both parts does not have a general analytical solution.

However, if only a 'small' step $h$ is taken along $z$, then the two parts can be treated separately with only a 'small' numerical error. One can therefore first take a small nonlinear step,

$A_{N}(t,z+h)=\exp \left[i\gamma |A|^{2}h\right]A(t,z),$

using the analytical solution.

The dispersion step has an analytical solution in the frequency domain, so it is first necessary to Fourier transform $A_{D}$ using

${\tilde {A}}_{D}(\omega ,z)=\int _{-\infty }^{\infty }A_{D}(t,z)\exp[i(\omega -\omega _{0})t]dt$,

where $\omega _{0}$ is the center frequency of the pulse. It can be shown that using the above definition of the Fourier transform, the analytical solution to the linear step is

${\tilde {A}}(\omega ,z+h)=\exp \left[{i\beta _{2} \over 2}(\omega -\omega _{0})^{2}h\right]{\tilde {A}}_{D}(\omega ,z).$

By taking the inverse Fourier transform of ${\tilde {A}}(\omega ,z+h)$ one obtains $A\left(t,z+h\right)$; the pulse has thus been propagated a small step $h$. By repeating the above $N$ times, the pulse can be propagated over a length of $Nh$.

The Fourier transforms of this algorithm can be computed relatively fast using the fast Fourier transform (FFT). The split-step Fourier method can therefore be much faster than typical finite difference methods.

External references

• Thomas E. Murphy, Software, http://www.photonics.umd.edu/software/ssprop/

• Andrés A. Rieznik, Software, http://photonics.incubadora.fapesp.br

• Prof. G. Agrawal, Software, http://www.optics.rochester.edu/workgroups/agrawal/grouphomepage.php?pageid=software

• Thomas Schreiber, Software, http://www.fiberdesk.com