How does split step Fourier method help four wave mixing?

[eahaidar]

Just a question
How does solving the nonlinear Schrödinger equation using split step Fourier method makes us understand the four wave mixing process in optical fiber ?
Any examples on how that happens
Thank you

 

[soarce]

The spit-step Fourier method is a numerical method for solving time dependent differential equations. It is just a tool, it may helps you understand / study time dependent phenomena which are not easily deduced from the equations and their initial conditions. For instance, this method is used to study the dynamics of nonlinear waves (e.g. solitons) in two- and three-dimensional scenarios. In general, one may obtain some analitic results which may predict some features of the dynamics, but in the end numerical simulations are used to confirm the analitic results.

 

[eahaidar]

The spit-step Fourier method is a numerical method for solving time dependent differential equations. It is just a tool, it may helps you understand / study time dependent phenomena which are not easily deduced from the equations and their initial conditions. For instance, this method is used to study the dynamics of nonlinear waves (e.g. solitons) in two- and three-dimensional scenarios. In general, one may obtain some analitic results which may predict some features of the dynamics, but in the end numerical simulations are used to confirm the analitic results.

I always thought it would help understand how one optical light wave will propagate
My question is how solving this question problem by split step will show four wave mixing results which relies on more than one light wave ?how would it show the idler the signal
Is there any work done on that ?
Any code any example I can rely on?
Thank you for your time

 

[soarce]

There are many studies regarding the four-wave mixing process. First, one should establish the model, i.e. the equations, to be simulated numerically. Depending on the effects which one wants to study some modell may be more appropiate in comparison with others.
For instance, in this paper http://古河電工.jp/review/fr019/fr19_12.pdf the model is fairly simple: propagation along z axis and nonlinear coupling, it has no dispersion term. In general, in fiber optics the transverse effects (difraction) are neglected. In this case maybe the Fourier split-step method may be inappropiate.

Are you familiar with the Fourier split-step method or why did you choose this particular method ?

 

[eahaidar]

I am working with four wave mixing and I really want to know how to simulate it using the split step Fourier theory
That's why

 

[soarce]

The Fourier split-step method is based on splitting the time evolution operator into linear and nonlinear components: the linear part can be implemented by Fourier transform (e.g. diffraction or dispersion terms) while the nolinear parts can be approximated by a multiplication factor.

An example can be found here: http://en.wikipedia.org/wiki/Split-step_method
What are the relevant equations of your four wave mixing process?