Commit 31454bad authored by Mathieu RASSON's avatar Mathieu RASSON

Report equations and diffraction code

parent 60ce0b29
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......@@ -63,10 +63,13 @@ 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)
\subsection{Model of Filamentation}
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*}
\begin{equation}
\mathbf{E}(r,\theta,z,t) = \bm{\epsilon} \, \mathcal{E}(z,r,t) \, e^{-i(\omega_0 t - k_0 z)}
\label{eq:form}
\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}
......@@ -94,9 +97,101 @@ A delayed Kerr effect can as well be introduced as
+ i k_0 n_2 \left[\int_{-\infty}^{+\infty} {\rm d}t' R(t-t') |\mathcal{E}(t')|^2 \right] \mathcal{E}
\end{equation*}
Defocusing effects are all linked with the plasma created by ionisation. The laser pulse is intense enough to free electrons from the potential of cores. The first source of ionisation is due to multiphoton absorption, which adds the energies of several photons in order to reach the ionisation energy. The contribution on the field can be written
\begin{equation*}
- \dfrac{\beta^{(K)}}{2} |\mathcal{E}|^{2K - 2} \mathcal{E}
\end{equation*}
where $K$ is the number of photons absorbed at a time, $K = 7$ for air, and $\beta^{(K)}$ accounts for the efficiency of this absorption.
The second defocusing effects is due to the reaction of charges in the plasma to the excitation of the electric field. Considering a Drude model for electrons, their answer to a periodic excitation creates a delay in their polarisation. This answer reads
\begin{equation*}
- \dfrac{\sigma}{2} (1 + i \omega_0 \tau) \rho\mathcal{E}
\end{equation*}
for the field, where $\sigma$ is the efficiency of the interaction and $\tau$ the mean fly time of an electron.
% Equations for the density of charges
As a plasma is created by the electric field, the density of charges changes and is coupled to the evolution of the amplitude of the pulse. First, multiphoton absorption contributes to plasma creation as
\begin{equation*}
\dfrac{\beta^{(K)} |\mathcal{E}|^{2K}}{K \hbar \omega}
\end{equation*}
Then, freed electrons can ionise bounded electrons after a collision according to
\begin{equation*}
\dfrac{1}{n_b} \dfrac{\sigma}{E_g} \rho |\mathcal{E}|^2
\end{equation*}
term which is therefore proportional to $\rho$, density of free charges. And, plasma creation competes with electrons recombination into atoms which decreases the density of free charges in the extent of
\begin{equation*}
a \rho^2
\end{equation*}
where $a$ accounts for the efficiency of the process.
Thus, filamentation can be described by to non linear and coupled differential equations. First, the evolution of the amplitude of the laser pulse emitted
\begin{equation}
\begin{split}
\dfrac{\partial \mathcal{E}}{\partial z} \quad = \quad
&\dfrac{i}{2k_0} \left( \dfrac{\partial^2}{\partial r^2} + \dfrac{1}{r} \dfrac{\partial}{\partial r} \right) \mathcal{E} \\
&- \dfrac{i k_0^{(2)}}{2} \dfrac{\partial^2 \mathcal{E}}{\partial t^2} \\
&+ i k_0 (1-f) n_2 |\mathcal{E}|^2 \mathcal{E} + i k_0 f n_2 \left[\int_{-\infty}^{+\infty} {\rm d}t' R(t-t') |\mathcal{E}(t')|^2 \right] \mathcal{E} \\
&- \dfrac{\beta^{(K)}}{2} |\mathcal{E}|^{2K - 2} \mathcal{E} - \dfrac{\sigma}{2} (1 + i \omega_0 \tau) \rho\mathcal{E}
\end{split}
\label{eq:field}
\end{equation}
and, the evolution of the density of free charges through plasma creation
\begin{equation}
\dfrac{\partial \rho}{\partial t} \quad = \quad \dfrac{\beta^{(K)} |\mathcal{E}|^{2K}}{K \hbar \omega} + \dfrac{1}{n_b} \dfrac{\sigma}{E_g} \rho |\mathcal{E}|^2 - a \rho^2
\label{eq:charge}
\end{equation}
\subsection{Numerical Integration Strategy}
\subsubsection{Cylindrical Diffraction}
The set of equations considered is to be solved step by step. First, the evolution of the field, non coupled with the density of charges is considered. Autofocusing and defocusing effects in which the density of charges does not intervene can create a first form of filamentation.
The numerical method at the heart of the solver used is a Finite Differences Method, with a Crank-Nicholson scheme. As a symmetric combination of a explicit and implicit schemes, it is an order 2 scheme in all the dimensions of the problem.
This method is first to be tested on the cylindrical diffraction part of equation \ref{eq:field}. A two-dimensional advection-diffusion equation, $r$ diffusion along $z$, is considered, where the advection coefficient depends on $r$ because of the rotation invariance. Consequently the adapted Crank-Nicholson scheme reads
\begin{equation}
\begin{split}
\dfrac{E_j^{n+1} - E_j^n}{\Delta z} \quad = \quad & \dfrac{D}{2 \Delta r^2} ( E_{j+1}^{n} - 2 E_{j}^{n} + E_{j-1}^{n} + E_{j+1}^{n+1} - 2 E_{j}^{n+1} + E_{j-1}^{n+1} ) \\
&+ \dfrac{V(r_j)}{4 \Delta r} ( E_{j+1}^{n} - E_{j-1}^{n} + E_{j+1}^{n+1} - E_{j-1}^{n+1} ) \\
\end{split}
\end{equation}
where the discretised amplitude is $E_j^n = \mathcal{E}(z_n, r_j)$, the constant diffusion coefficient is $D = i / 2k_0$ and the advection coefficient is
\begin{equation*}
V(r) = \dfrac{i}{2 k_0} \dfrac{1}{r} \, \mathbb{I}_{r >0}
\end{equation*}
with $\mathbb{I}_{r >0}$ to avoid a singularity at $r=0$. This precaution does not impact the physical behaviour of the scheme thanks to the border conditions considered. The rotation invariance situation and a decreasing field at infinity can be written
\begin{equation*}
\begin{cases}
\mathcal{E}(r \to +\infty) = 0 \\
\partial_r \mathcal{E} (r=0) = 0
\end{cases}
\end{equation*}
Because of the null gradient condition, the advection coefficient at $r=0$ does not intervene.
Thus, with the Crank-Nicholson scheme and border conditions known, an initial Gaussian beam can be set at $z=0$,
\begin{equation*}
\mathcal{E}(z=0, r) = \sqrt{\dfrac{2 \mathcal{P}_{\rm in}}{\pi w_0^2}} \, \exp \left( -\dfrac{r^2}{w_0^2} \right)
\end{equation*}
to test the cylindrical diffraction solver. Here $\mathcal{P}_{\rm in}$ is the power of the initial laser pulse and $w_0$ the initial waist of the Gaussian beam.
\subsubsection{Time Dispersion}
% Double steps description
\subsubsection{Non Linear Terms}
% Multiple steps description to symmetrise non linear terms
\newpage
%----------------Results and Analysis-------------------------
% Results, figures, plots, convergence issues and precision
......
......@@ -8,7 +8,7 @@
%Symboles de maths, couleurs, degré
\usepackage{color,textcomp,amsmath,bm}
%Fonte
\usepackage[utopia]{mathdesign}
\usepackage[utopia, cal=cmcal]{mathdesign}
%Mise en page
\usepackage[top=3cm , bottom=2.5cm , left=3cm , right=3cm,%
headheight=15pt,footskip=1cm]{geometry}
......
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