Commit 60ce0b29 authored by Mathieu RASSON's avatar Mathieu RASSON

Report beginning of equations

parent b75ae6b4
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......@@ -63,6 +63,37 @@ Finite Differences and Crank-Nicholson scheme are the core of the numerical meth
% Approximations, equations, terms and significations
% Numerical steps, each introduction method (CN, half-step, non linear multi-step)
The propagation of the electric field is described thanks to Maxwell equations written for a slow varying envelope. A field with a fixed polarisation $\bm{\epsilon}$ is considered as
\begin{equation*}
\mathbf{E}(r,\theta,z,t) = \bm{\epsilon} \, \mathcal{E}(r,z,t) \, e^{-i(\omega_0 t - k_0 z)}
\end{equation*}
with an rotation invariant envelope $\mathcal{E}$, and a carrier defined by $k_0 = \omega_0 / c$. The main shape of the envelope is a short Gaussian pulse leading to the intensities required for filamentation. Those choices of approximations explain the form of the diffraction part of the differential problem.
\begin{equation*}
\dfrac{\partial \mathcal{E}}{\partial z} = \dfrac{i}{2k_0} \left( \dfrac{\partial^2}{\partial r^2} + \dfrac{1}{r} \dfrac{\partial}{\partial r} \right) \mathcal{E}
\end{equation*}
Other effects taken into account are added term by term. First, the dispersion of air is considered through the development of the wave vector
\begin{equation*}
k(\omega) = k_0 + k_0^{(1)} (\omega - \omega_0) + k_0^{(2)} (\omega - \omega_0)^2
\end{equation*}
The first order leads to the definition of the group speed of the pulse $v_g = 1 / k_0^{(1)}$. The second order of the development accounts for differences of propagation speed between the harmonics of the pulse. With dispersion the time evolution of the beam can be introduced. The time differentiation of the total field in Maxwell leads to two terms for the envelope. The first differentiation of the envelope is discarded through the referential change
\begin{equation*}
t = t_{\rm lab} - \dfrac{z}{v_g}
\end{equation*}
from the referential of the laboratory to that of the pulse. In this new referential remains the time dispersion of the envelope.
\begin{equation*}
- \dfrac{i k_0^{(2)}}{2} \dfrac{\partial^2 \mathcal{E}}{\partial t^2}
\end{equation*}
Then the Kerr effect is introduced. It consists in a modification of the refraction index of the air because of the intensity of the beam itself. If $\mathcal{I}$ is the intensity of the pulse, the varying index reads $n(\mathcal{I}) = n_0 + n_2 \mathcal{I}$, where $n_0$ is close to $1$. As the index increases with intensity, the beam focuses, which increases its intensity and it focuses again. There is theoretically a collapse if the defocusing effects which follow are not considered. Therefore the instantaneous Kerr effect added to the propagation reads
\begin{equation*}
+ i k_0 n_2 |\mathcal{E}|^2 \mathcal{E}
\end{equation*}
A delayed Kerr effect can as well be introduced as
\begin{equation*}
+ i k_0 n_2 \left[\int_{-\infty}^{+\infty} {\rm d}t' R(t-t') |\mathcal{E}(t')|^2 \right] \mathcal{E}
\end{equation*}
\newpage
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\usepackage[english]{babel}
\usepackage[autolanguage]{numprint} %Format nombre
%Symboles de maths, couleurs, degré
\usepackage{color,textcomp,amsmath}
\usepackage{color,textcomp,amsmath,bm}
%Fonte
\usepackage[utopia]{mathdesign}
%Mise en page
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