This is an abstract of our project. The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is.

Filamentation is a non linear mode of propagation of an intense laser beam created by the equilibrium between autofocusing and defocusing effects. This balanced evolution results in a propagation of the beam on rather long distances without diffusion of the initial power in all directions. The initial spatial coherence in preserved.

The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is.

The main purpose of this project is to illustrate this non linear behaviour thanks to numerical integration methods. Maxwell equations ruling the electric field and plasma creation are at the heart of the physics of this phenomenon. Considering a cylindrical beam, a three-dimensional solver of those equations is implemented in order to explicit the dynamic evolution of the intense electric field.

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This is an introduction to our project. The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is.

Filamentation can be accounted for thanks to a classical treatment of light through Maxwell equations, coupled with charge density evolution. As a non linear effect, it is characterised by the high intensities considered, leading to a Kerr effect and ionisation of air. Those are the two main terms of the equilibrium underneath filamentation: the Kerr effect is responsible for the beam's autofocusing, whereas the plasma created adds a diffusion term which defocuses the beam.

The combination of effects at stake explains the numerous terms added into Maxwell equations, beside the native diffraction of the beam. To describe the propagation of the envelope of the field, carrying its energy, partial derivative equations are considered, coupling time and space. As a four-dimensional numerical integration would be heavy to run, a cylindrical beam is hear chosen in order to focus on the dynamic evolution of the beam. A three-dimensional scheme is to be implemented.

Finite Differences and Crank-Nicholson scheme are the core of the numerical methods chosen to integrate the differential problem. Their implementation is done step by step, going from cylindrical diffraction to complete filamentation, adding a new effect at each step. The idea is to check the behaviour of the algorithm implemented according to predictable results. The last step is the analysis of the properties of a filamentory propagation.

The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is.

\section{Equations and Numerical Treatment}

% Approximations, equations, terms and significations

% Numerical steps, each introduction method (CN, half-step, non linear multi-step)

This is the first section of our project. The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is.

\section{Conclusion}

The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is. The crucial point about this project is to remember how exceptional it is.