Commit 65223665 by Mathieu RASSON

 ... ... @@ -229,11 +229,11 @@ With this method all non linear terms can be added one by one, to be tested indi \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 evolution of the \textbf{electric field} and that of the \textbf{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 $\rho^2$. The equation ruling the creation of free charges is treated in a \textbf{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*} ... ... @@ -245,11 +245,11 @@ Then the first integral can be discretised regarding a time step $\Delta t$ and & + \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. As the field is known, the charge density can be calculated from \textbf{null initial conditions} 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. Thus, all effects leading to the \textbf{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. \newpage ... ... @@ -289,6 +289,9 @@ Thus, all effects leading to the saturation of the intensity autofocused, to its % Behaviour up to 10 P_cr % Multifilamentation? % Analytical equilibrium between Kerr and Plasma % Saturation intensity independent on P_in? \newpage ... ... @@ -307,10 +310,22 @@ Thus, all effects leading to the saturation of the intensity autofocused, to its \appendix %---------------Users Guide----------------- %---------------User's Guide----------------- % How to use codes and reproduce results \section{Users Guide} \section{User's Guide} The different Crank-Nicholson solvers, according to the effects implemented, are in classes saved in \texttt{.py} files at \texttt{/codes/CN\_classes/}. The names refer to the step towards full filamentation: \begin{itemize} \item \texttt{CCCN.py}: Cylindrical Diffraction; \item \texttt{HSCN.py}: Diffraction and Dispersion; \item \texttt{KCN.py}: Kerr effect; \item \texttt{FCN.py}: Full Filamentation. \end{itemize} Each of those classes are accompanied by a Jupyter Notebook named \texttt{*\_script.ipynb}, which instantiates the solver and runs the integration. Then, for linear integrations, data are saved in a \texttt{.npy} file and analysed in another Jupyter Notebook at \texttt{/codes/analyse}. For non linear integrations, results are analysed directly in the integration notebook. \newpage ... ...