Commit 5c173a45 authored by Mathieu RASSON's avatar Mathieu RASSON

Journal and Report update

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\title{Laser Filamentation}
\author{Mathieu Rasson \and Antoine Ravetta}
\date{November and December 2019}
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\makeatletter
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......@@ -71,8 +71,6 @@ The propagation of a high intensity laser pulse in air is considered. As the int
\section*{November 5th 2019}
%Beginning of the project. Organisation. Physics approach of coupled non linear propagation equations. Terms elaborated. Coding steps listed. Work division and documents sharing (with coordinators) to be done.
Course about versions management with a \texttt{git} tool. Delocalised recovery of files on \texttt{Gitlab} provided by the BR. \textbf{Warning}: give access to Couairon and Ferrero.
\subsubsection*{Physics of the Problem}
......@@ -222,6 +220,23 @@ Non linear algorithm does end with Kerr effect. Creation of a concentration poin
\item Add defocusing terms.
\end{itemize}
\section*{Tuesday 26th}
Energy check performed with all configurations until Kerr effect. Collapse distance compared with Marburger's law. Defocusing terms added with coupling with charge density (theory: eq.~118 p.~34, implementation: p.~56).
\subsection*{To Be Done}
\begin{itemize}
\item Check diffraction with complete solver;
\item Propagation animation;
\item Filamentation analysis: choose integration steps coherent with physical effects. Too large steps can create divergences: $z$, $r$ and $t$ scales of non linear phenomena;
\item Illustrate effects from $\mathcal{P}_{\rm crit}$ to $10 \, \mathcal{P}_{\rm crit}$ with a potential multifilamentation from $5 \, \mathcal{P}_{\rm crit}$;
\item Try other parameters: medium (dispersion, Kerr effect\dots)
\item Quantitative analysis with $\max\left( |\mathcal{E}|^2 \right)$ along $z$ and track reflections with a log scale;
\item \textbf{Report}: highlight important points, add precise quotations.
\end{itemize}
\end{multicols}
\end{document}
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......@@ -176,7 +176,7 @@ to test the cylindrical diffraction solver. Here $\mathcal{P}_{\rm in}$ is the p
\subsubsection{Time Dispersion}
In the referential of the laser pulse, taking into account the \textbf{dispersion} of the medium consists in considering the \textbf{time evolution} of the amplitude of the beam. Therefore, a third dimension is added to our scheme, which is treated thanks to a \textbf{double step method}.
In the referential of the laser pulse, taking into account the \textbf{dispersion} of the medium consists in considering the \textbf{time evolution} of the amplitude of the beam. Therefore, a third dimension is added to our scheme, which is treated thanks to a \textbf{double step method}. The field is discretised as $(n,j,k) \mapsto \mathcal{E}(z_n, r_j, t_k)$.
The differential problem considered so far, with cylindrical diffraction and dispersion, can be represented as follows
\begin{equation*}
......@@ -205,7 +205,7 @@ Because of the referential change presented before, the initial conditions are i
\end{equation*}
where $t_p$ is the time extension of the pulse.
% Gaussian normalisation ?
% Gaussian normalisation (bidimensional)?
\subsubsection{Non Linear Terms}
......@@ -227,7 +227,29 @@ The non linear terms are added to the $t$ operator regarding its simplicity comp
With this method all non linear terms can be added one by one, to be tested individually. With Kerr effect appears a collapse of the laser beam and multiphotons absorption enables laser filamentation.
% Coupling with the density of charges
\subsubsection{Plasma Coupling}
The evolution of the electric field and that of the density of free charges are coupled through ionisation and the answer of those charges under the laser pulse. As the electric field, this density of free charges is discretised according to $(n,j,k) \mapsto \rho(z_n, r_j, t_k)$.
The field and the density has to be estimated together at each propagation step to account for the coupling. At each $z$ step, the electric field is calculated first thanks to the charge density at the $z_{n-1}$, and the field calculated makes it possible to update the value of the density at $z_n$. Then there is only an integration along $t$ for all $r$ position.
The equation ruling the creation of free charges is treated in a semi-analytical manner in time. This method avoids any negative value which could be created by an integration tool as \texttt{odeint} from \texttt{scipy}, if wrong steps are chosen. The idea is to write a formal integration of the first order linear equation leading $\rho$, neglecting recombination in $\rho^2$.
\begin{equation*}
\rho(t) = \int_{-\infty}^t \dfrac{\beta^{(K)}}{\hbar \omega_0 K} |\mathcal{E}|^{2K} (t') \int_{t'}^t \dfrac{\sigma}{n_b^2 E_g} |\mathcal{E}|^{2}(t'') \, {\rm d}t' {\rm d}t''
\end{equation*}
Then the first integral can be discretised regarding a time step $\Delta t$ and the second one approximated by a trapeze method. The density $\rho(t + \Delta t)$ can thus be written according to $\rho(t)$ as
\begin{equation*}
\begin{split}
\rho(z_{n+1}, r_j, t_{k+1}) \quad = \quad & \exp \left\lbrace \dfrac{\sigma}{n_b^2 E_g} \dfrac{1}{2} \left(|\mathcal{E}|^{2}(z_{n+1}, r_j, t_{k+1}) + |\mathcal{E}|^{2}(z_{n+1}, r_j, t_{k} \right) \right\rbrace \\
& \times \left( \rho(z_{n+1}, r_j, t_{k}) + \dfrac{\Delta t}{2} \dfrac{\beta^{(K)}}{\hbar \omega_0 K} |\mathcal{E}|^{2K}(z_{n+1}, r_j, t_{k}) \right) \\
& + \dfrac{\Delta t}{2} \dfrac{\beta^{(K)}}{\hbar \omega_0 K} |\mathcal{E}|^{2K}(z_{n+1}, r_j, t_{k+1})
\end{split}
\end{equation*}
As the field is known, the charge density can be calculated from a null initial condition at the beginning of the time integration box.
Then, the coupling of field and charge density through the reaction of free charges is inserted with the non linear terms in the Crank-Nicholson scheme. The same symmetrisation is applied to this effect.
Thus, all effects leading to the saturation of the intensity autofocused, to its decrease by defocusing and to autofocusing again are taken into account is the propagation solver implemented. Now integration parameters, such as the dimensions of the box and integrations steps, have to be adapted so that the numerical algorithm reproduces physical phenomena.
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\title{Laser Filamentation}
\author{\textsc{Rasson} Mathieu \and \textsc{Ravetta} Antoine}
\date{November to December 2019} %inutile
\date{November to December 2019} % inutile
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