diff --git a/Dispersion_Relation.tex b/Dispersion_Relation.tex index af9c7ac..b9d9f7f 100644 --- a/Dispersion_Relation.tex +++ b/Dispersion_Relation.tex @@ -1,524 +1,495 @@ -%% LyX 2.2.1 created this file. For more info, see http://www.lyx.org/. -%% Do not edit unless you really know what you are doing. -\documentclass[french]{article} -\usepackage[T1]{fontenc} -\usepackage[latin9]{inputenc} -\usepackage{amsmath} -\usepackage{amssymb} -\usepackage{graphicx} -\usepackage{babel} -\makeatletter -\addto\extrasfrench{% - \providecommand{\og}{\leavevmode\flqq~}% - \providecommand{\fg}{\ifdim\lastskip>\z@\unskip\fi~\frqq}% -} - -\makeatother -\begin{document} - -\section{\label{sec:annexe_relation_dispersion}Relation de dispersion} - -\subsection{Expressions de la relation de dispersion $\mathbf{k}(\omega)$} - -\subsubsection{Expression en fonction de $n^{2}$} - -On rappelle que dans un milieu diélectrique non-dispersif\footnote{La \emph{dispersion} est le phénomène affectant une onde dans un milieu -\emph{dispersif}, c'est-à-dire dans lequel les différentes fréquences -constituant l'onde ne se propagent pas à la même vitesse.}, isotrope, homogène et sans perte, la direction du vecteur d'onde -$\mathbf{k}=k\mathbf{\hat{k}}$ est donnée par -\begin{equation} -\mathbf{\hat{k}}\cdot\mathbf{E}=0\;\;\mathbf{H}=\frac{n}{Z_{0}}\hat{\mathbf{k}}\times\mathbf{E}\label{eq:vecteur_onde_vide} -\end{equation} -où $k=nk_{0}$ est le nombre d'onde, $\mathbf{\hat{k}}$ est le vecteur -unitaire dans la direction du vecteur d'onde $\mathbf{k}$, $n=\sqrt{\varepsilon_{r}}$ -l'indice optique du milieu et $Z_{0}=\sqrt{\frac{\mu_{0}}{\varepsilon_{0}}}=\mu_{0}c_{0}=\frac{1}{\varepsilon_{0}c_{0}}$ -l'impédance du vide. Ainsi, le trièdre $\mathbf{E,}\mathbf{H,\hat{k}}$ -est direct. L'équation d'onde dans un tel milieu est -\begin{equation} -\mathbf{k}\times\mathbf{k}\times\tilde{\mathbf{E}}+k^{2}\tilde{\mathbf{E}}=\mathbf{0} -\end{equation} -La condition pour que ce système de trois équations et trois inconnues -$(\tilde{E}_{x},\tilde{E}_{y},\tilde{E}_{z})$ aient une solution, -conduit à résoudre la \emph{relation de dispersion }entre le vecteur -d'onde $\mathbf{k}$ et la fréquence $\omega$, i.e.$\mathbf{k}(\omega)$. -Dans ce milieu, cette relation est simple : $k=\sqrt{\mu\varepsilon}\omega=\frac{c}{n}\omega$\cite[(7.4)]{Jackson1998}\footnote{Dans un matériau avec pertes $k^{2}=\mu\varepsilon\omega^{2}-j\omega\mu\sigma$\cite[§8.2]{Bladel2007}.}. -Dans un plasma froid magnétisé, la situation est radicalement différente, -car les courants de polarisation générés par les mouvements électroniques -et ioniques modifient la polarisation des ondes planes ainsi que leur -dispersion. Différentes branches de dispersion, ou \emph{modes}, apparaissent\cite[chap.8]{Rax2005}. - -On rappelle l'expression des équations de Maxwell en régime harmonique -pour une onde plane de vecteur d'onde $\mathbf{k}$ : - -\begin{eqnarray} -\mathbf{k}\times\mathbf{\tilde{E}} & = & \omega\mu_{0}\mathbf{\tilde{H}}\\ -\mathbf{k}\times\mathbf{\tilde{H}} & \mathbf{=} & -\omega\varepsilon_{0}\mathbb{K}\cdot\mathbf{\tilde{E}} -\end{eqnarray} -Pour déterminer les propriétés de ces modes, on étudie les solutions -de l'\emph{équation d'onde} déduite des deux précédentes équations\footnote{NB : L'équation d'onde ne dépend pas de la convention temporelle choisie.} -: - -\begin{equation} -\mathbf{n}\times\mathbf{n}\times\mathbf{\tilde{E}}+\mathbb{K}\cdot\mathbf{\tilde{E}}=\mathbf{0}\label{eq:Helmoltz} -\end{equation} -où $\mathbf{n}=\mathbf{k}/k_{0}$ correspond au vecteur d'indice de -réfraction, dont la direction est celle du vecteur d'onde $\mathbf{k}$ -et l'amplitude celle de l'indice de réfraction. A priori, la matrice -$\mathbb{K}$ dépend des trois composantes du vecteur d'onde $\mathbf{k}$. -En choisissant le système de coordonnées cartésien l'équation \ref{eq:Helmoltz} -peut s'écrire sous forme matricielle\footnote{On peut s'économiser un peu de calcul vectoriel grâce à un peu d'algèbre -et le formalisme des dyadiques\cite{Belov2003,Lindell1995}. En utilisant -l'identité ``BAC-CAB'', le double produit vectoriel s'écrit $\mathbf{n}\times\mathbf{n}\times\mathbf{\tilde{E}}=\mathbf{n}\left(\mathbf{n}\cdot\mathbf{\tilde{E}}\right)-\mathbf{\tilde{E}}\left(\mathbf{n}\cdot\mathbf{n}\right)=\mathbf{n}\left(\mathbf{n}\cdot\mathbf{\tilde{E}}\right)-n^{2}\mathbf{\tilde{E}}$. -Avec l'opération dyadique $\mathbf{n}\left(\mathbf{n}\cdot\tilde{\mathbf{E}}\right)=\mathbf{nn}\cdot\mathbf{\tilde{E}}$ -et $\mathbb{I}=\mathbf{xx}+\mathbf{yy}=\mathbf{zz}$ l'opérateur dyadique -unité vérifiant $\mathbf{\tilde{E}}=\mathbb{I}\cdot\tilde{\mathbf{E}}$on -peut alors factoriser l'équation \ref{eq:Helmoltz} en un opérateur -dyadique $\left(\mathbf{nn}-n^{2}\mathbb{I}+\mathbb{K}\right)\cdot\tilde{\mathbf{E}}=\mathbf{0}$ -qui donne directement l'expression matricielle de l'équation \ref{eq:relation_disp_matr}.} : -\begin{equation} -\left(\begin{array}{ccc} -K_{xx}-n_{y}^{2}-n_{z}^{2} & K_{xy}+n_{x}n_{y} & K_{xz}+n_{x}n_{z}\\ -K_{yx}+n_{x}n_{y} & K_{yy}-n_{x}^{2}-n_{z}^{2} & K_{yz}+n_{y}n_{z}\\ -K_{zx}+n_{x}n_{z} & K_{zy}+n_{y}n_{z} & K_{zz}-n_{x}^{2}-n_{y}^{2} -\end{array}\right)\left(\begin{array}{c} -\tilde{E_{x}}\\ -\tilde{E}_{y}\\ -\tilde{E}_{z} -\end{array}\right)=\mathbf{0}\label{eq:relation_disp_matr} -\end{equation} - Si l'on suppose en plus que le vecteur d'onde $\mathbf{k}$ (ie. -la propagation de l'onde) soit contenu dans le plan $x-z$ (ie $k_{y}=n_{y}=0$), -l'équation précédente devient : - -\begin{equation} -\left(\begin{array}{ccc} -K_{xx}-n_{z}^{2} & K_{xy} & K_{xz}+n_{x}n_{z}\\ -K_{yx} & K_{yy}-n_{x}^{2}-n_{z}^{2} & K_{yz}\\ -K_{zx}+n_{x}n_{z} & K_{zy} & K_{zz}-n_{x}^{2} -\end{array}\right)\left(\begin{array}{c} -\tilde{E_{x}}\\ -\tilde{E}_{y}\\ -\tilde{E}_{z} -\end{array}\right)=\mathbf{0}\label{eq:relation_disp_matr_full} -\end{equation} -Si le plasma est homogène (indépendant de $\mathbf{r}$) on peut exploiter -l'équivalence entre toutes les directions perpendiculaires au champ -magnétique statique pour prédire que $\mathbb{K}$ doit être fonction -de $k_{\parallel}$et $k_{\perp}^{2}$ seulement. - -\begin{figure} -\begin{centering} -\includegraphics[width=0.5\textwidth]{figures/geometrie_dispersion_plasma} -\par\end{centering} -\caption{Géométrie cartésienne du milieu plasma.\label{fig:G=0000E9om=0000E9trie-cart=0000E9sienne-plasma}} -\end{figure} - -Si $\mathbb{K}$ est le tenseur de permittivité d'un plasma froid -(\ref{eq:tenseur_stix}) définit dans l'annexe \ref{sec:Tenseur-de-permittivit=0000E9}, -c'est-à-dire tel que $z$ soit parallèle au champ magnétique (Figure -\ref{fig:G=0000E9om=0000E9trie-cart=0000E9sienne-plasma}), alors -on a: - -\begin{equation} -\left(\begin{array}{ccc} -S-n_{z}^{2} & jD & n_{x}n_{z}\\ --jD & S-n_{x}^{2}-n_{z}^{2} & 0\\ -n_{x}n_{z} & 0 & P-n_{x}^{2} -\end{array}\right)\left(\begin{array}{c} -\tilde{E_{x}}\\ -\tilde{E}_{y}\\ -\tilde{E}_{z} -\end{array}\right)=\mathbf{0}\label{eq:relation_disp_matr_froid} -\end{equation} -On définit $\theta$ comme l'angle entre le vecteur d'onde $\mathbf{k}$ -et la direction $\hat{\mathbf{e}}_{z}$ du champ magnétique, soit -$n_{x}=n_{\perp}=n\sin\theta$ et $n_{z}=n_{\parallel}=n\cos\theta$, -on a alors \footnote{Pour obtenir la version harmonique en convention $-j\omega t$, il -faut remplacer $n^{2}\rightarrow-n^{2}$ et $j\rightarrow-j$.}: - -\begin{equation} -\left(\begin{array}{ccc} -S-n^{2}\cos^{2}\theta & jD & n^{2}\cos\theta\sin\theta\\ --jD & S-n^{2} & 0\\ -n^{2}\cos\theta\sin\theta & 0 & P-n^{2}\sin^{2}\theta -\end{array}\right)\left(\begin{array}{c} -\tilde{E}_{x}\\ -\tilde{E}_{y}\\ -\tilde{E}_{z} -\end{array}\right)=\mathbf{0}\label{eq:cold_plasma_dispersion_relation_matrix} -\end{equation} - -L'existence de solutions non triviales à l'équation d'onde (les modes) -(\ref{eq:relation_disp_matr_full}) nécessite que le déterminant de -la matrice soit nul. Cette condition donne la \emph{relation de dispersion}, -qui pour un plasma froid peut s'écrire \cite[p.8-9]{Stix1992}\cite[§2.1.3]{Swanson2003}\cite[§18.1]{Brambilla1998}: - -\begin{equation} -\boxed{An^{4}-Bn^{2}+C=0}\label{eq:cold_plasma_dispersion_relation_n} -\end{equation} -avec\footnote{Où l'on a remarqué que $S^{2}-D^{2}=RL$. Les expressions $A,B,C$ -ne dépendent pas de la convention temporelle choisie. }: - -\begin{eqnarray} -A & = & S\sin^{2}\theta+P\cos^{2}\theta\\ -B & = & \left(S^{2}-D^{2}\right)\sin^{2}\theta+PS\left(1+\cos^{2}\theta\right)=RL\sin^{2}\theta+PS\left(1+\cos^{2}\theta\right)\\ -C & = & P\left(S^{2}-D^{2}\right)=PRL -\end{eqnarray} -Soit $n=n(\theta,\omega)=n(\hat{\mathbf{k}},\omega)$ la solution -de la relation de dispersion pour une fréquence $\omega$ et une direction -de propagation $\hat{\mathbf{k}}$ (ie. $\theta$) données. Une onde -plane d'indice $n$ et de nombre d'onde $\mathbf{k}=n\frac{\omega}{c}\mathbf{\hat{k}}$ -peut se propager dans le plasma à la fréquence $\omega$ et dans la -direction du vecteur unitaire $\mathbf{\hat{k}}$ en l'absence de -sources extérieures (plus exactement avec des sources situées à l'infini). -Une telle onde est appelée \emph{onde caractéristique} ou \emph{mode -de propagation} (mode propre) du plasma. - -L'équation (\ref{eq:cold_plasma_dispersion_relation_n}) est une équation -du second degré en $n^{2}$ ayant pour solution : -\begin{equation} -n^{2}=\frac{B\pm\sqrt{\Delta}}{2A}\label{eq:solution_cold_plasma_dispersion_relation_n} -\end{equation} -où son déterminant $\Delta=B^{2}-4AC$ vaut : -\begin{equation} -\Delta=(RL-PS)^{2}\sin^{4}\theta+4P^{2}D^{2}\cos^{2}\theta\label{eq:cold_plasma_determinant_dispersion_relation_n} -\end{equation} - -Le déterminant (\ref{eq:cold_plasma_determinant_dispersion_relation_n}) -n'est jamais négatif, ce qui signifie que l'équation (\ref{eq:cold_plasma_dispersion_relation_n}) -a toujours deux solutions réelles et distinctes en $n^{2}$. Ainsi, -les ondes planes dans un plasma froid sont soit purement propagatives -($n^{2}>0$) soit purement évanescentes ($n^{2}<0$) ; les oscillations -amorties sont exclues. La transition entre ces deux régimes a lieu -aux coupures ($n=0$) et aux résonances ($n\to\infty$). - -Les deux racines peuvent être confondues lorsque le déterminant être -nul, dans les cas particulier suivant : -\begin{itemize} -\item En propagation parallèle, ie $\theta=0$, lorsque $P=0$ ; -\item En propagation perpendiculaire, ie $\theta=\pi/2$, lorsque $RL=PS$. -\end{itemize} -Dans le cadre de l'approximation froide définie par l\textquoteright absence -de \emph{dispersion spatiale}, c'est-à-dire lorsque les éléments du -tenseur $\mathbb{K}$ ne dépendent pas de l'indice de réfraction $\mathbf{n}$ -(ie. du vecteur d'onde $\mathbf{k}$), l'équation (\ref{eq:cold_plasma_dispersion_relation_n}) -est une simple équation quadratique en $n^{2}$. Il existe donc deux -solutions distinctes qui peuvent se propager dans le plasma ; un plasma -froid est donc un milieu \emph{biréfringent} (les ondes peuvent être -évanescentes ou non, selon les caractéristiques du plasma). Si on -avait introduit des effets thermiques, les éléments du tenseur de -permittivité dépendraient alors du vecteur d'onde et de nouveaux modes -apparaitraient\cite[§1.2.2]{Dumont2007}. - -\subsubsection{Expression en fonction de $\theta$.} - -L'équation de dispersion (\ref{eq:cold_plasma_dispersion_relation_n}) -peut être exprimée sous diverses formes équivalentes. La relation -de dispersion (\ref{eq:cold_plasma_dispersion_relation_n}) peut être -exprimée en fonction de l'angle $\theta$\cite[§18.2]{Brambilla1998}: - -\begin{equation} -\tan^{2}\theta=-\frac{P\left(n^{2}-R\right)\left(n^{2}-L\right)}{\left(Sn^{2}-RL\right)\left(n^{2}-P\right)}\label{eq:cold_plasma_dispersion_relation_theta} -\end{equation} -En résolvant pour $n^{2}$ on obtient un cas particulier des équations -d'Appleton-Hartree, et en particulier pour $D=0$ -\begin{equation} -n^{2}=\begin{cases} -\frac{PS}{S\sin^{2}\theta+P\cos^{2}\theta} & \mbox{extraordinary mode}\\ -S & \mbox{oridnary mode} -\end{cases}\label{eq:cold_plasma_dispersion_relation_solution_n} -\end{equation} - - -\subsubsection{Expression en fonction de $n_{\parallel}$.} - -Lorsque le nombre d'onde $n_{\parallel}=n_{z}$ est définit par des -conditions extérieures, comme la structure d'une antenne, et en supposant -que la propagation est contrainte au plan (xOz) ($n_{y}=0$), la relation -de dispersion peut être exprimée en fonction de $n_{\perp}^{2}=n_{x}^{2}=n^{2}-n_{\parallel}^{2}$. -Ainsi, le déterminant de (\ref{eq:relation_disp_matr_froid}) donne -une équations quadratique en $n_{\perp}^{2}$\cite[§18.2]{Brambilla1998}\footnote{En convention $-j\omega t$: $A_{1}n_{\perp}^{4}-B_{1}n_{\perp}^{2}+C_{1}=0$ }: - -\begin{equation} -A_{1}n_{\perp}^{4}+B_{1}n_{\perp}^{2}+C_{1}=0\label{eq:cold_plasma_dispersion_relation_n_perp} -\end{equation} -avec - -\begin{eqnarray} -A_{1} & = & S\\ -B_{1} & = & \left(P+S\right)\left(n_{\parallel}^{2}-S\right)+D^{2}=RL+PS-n_{\parallel}^{2}(P+S)\\ -C_{1} & = & P\left(\left(n_{\parallel}^{2}-S\right)^{2}-D^{2}\right)=P(n_{\parallel}^{2}-R)(n_{\parallel}^{2}-L) -\end{eqnarray} -où on rappelle que $R=S+D$ et $L=S-D$. Les coupures, qui correspondent -aux transitions entre propagation et évanescence, sont alors définies -pour $n_{\perp}=0$. Les résonances pour $n_{\perp}\to\infty$. Comme -précédemment, le déterminant de l'équation est toujours positif ou -nul, l'équation (\ref{eq:cold_plasma_dispersion_relation_n_perp}) -possède donc deux solutions, qui peuvent éventuellement être complexes -conjuguées selon les conditions (fréquence, densité, etc.). La solution -s'exprime par \cite[§15.9.2]{Friedberg2007}: - -\begin{equation} -n_{\perp}^{2}=-\frac{P}{2S}\left(D^{2}/P-S-n_{\parallel}^{2}\pm\sqrt{\left(D^{2}/P-S-n_{\parallel}^{2}\right)^{2}+\frac{4SD^{2}}{P}}\right)\label{eq:cold_plasma_solution_n_perp} -\end{equation} -La solution dont la valeur est la plus grande, c'est-à-dire pour laquelle -la valeur de la vitesse de phase perpendiculaire $v_{\perp}=\omega/k_{\perp}$ -sera la plus petite, correspond à la branche dite \emph{lente}, l'autre -branche étant la solution dite \emph{rapide}. - -\begin{figure}[h] -\begin{centering} -\includegraphics[width=0.9\textwidth]{figures/n_perp_vs_ne}\label{Flo:n_perp_vs_ne}\caption{Exemple de solutions de l'équation de dispersion pour $n_{\perp}^{2}$ -en fonction de la densité électronique $n_{e}$. \textbf{$n_{\parallel}=2.0$ -}et\textbf{ $B_{0}=2.95$~}T, f=3.7GHz.} -\par\end{centering} -\end{figure} - -Pour des valeurs de $\left|P\right|$ grande ($\omega\ll\omega_{pe}$), -les deux racines (\ref{eq:cold_plasma_solution_n_perp}) peuvent s'approcher -par au premier ordre en $\omega^{2}/\omega_{pe}^{2}$ (cf. \cite[p.222]{Brambilla1998}). -De la même façon, dans le voisinage des fréquences LH, on peut faire -l'hypothèse $D\approx0$. Dans ce cas, l'équation de dispersion s'écrit -(avec la convention temporelle $e^{j\omega t}$): -\begin{eqnarray} -A_{1}n_{\perp}^{4}+B_{1}n_{\perp}^{2}+C_{1} & = & 0\label{eq:cold_plasma_dispersion_relation_n_perp_approx1} -\end{eqnarray} -avec -\begin{eqnarray} -A_{1} & = & S\\ -B_{1} & = & S^{2}+PS-n_{\parallel}^{2}(P+S)\\ -C_{1} & = & P(n_{\parallel}^{2}-S)^{2} -\end{eqnarray} -Les solutions de l'équation de dispersion \ref{eq:cold_plasma_dispersion_relation_n_perp_approx1} -sont dans ce cas : -\begin{eqnarray} -n_{\perp}^{2}=n_{\perp,F}^{2} & = & -\left(S+n_{\parallel}^{2}\right)\label{eq:n_perp_fast_wave_solution_approximate}\\ -n_{\perp}^{2}=n_{\perp,S}^{2} & = & -\frac{P}{S}\left(S+n_{\parallel}^{2}\right)\label{eq:n_perp_slow_wave_solution_approximate} -\end{eqnarray} -Enfin, toujours au voisinage du domaine de plasma de bord, $S\approx1$ -d'où \ref{eq:cold_plasma_dispersion_relation_n_perp_approx1} : -\begin{equation} -n_{\perp}^{4}+\left(1-n_{\parallel}^{2}\right)\left(1+P\right)n_{\perp}^{2}+P\left(n_{\parallel}^{2}-1\right)^{2}=0 -\end{equation} -qui a pour solutions: -\begin{eqnarray} -n_{\perp}^{2}=n_{\perp,F}^{2} & = & -\left(1-n_{\parallel}^{2}\right)\label{eq:n_perp_fast_wave_solution_approximate2}\\ -n_{\perp}^{2}=n_{\perp,S}^{2} & = & -P\left(1-n_{\parallel}^{2}\right)\label{eq:n_perp_slow_wave_solution_approximate2} -\end{eqnarray} - - -\subsection{Polarisation des champs} - -Chaque mode propre de la solution de dispersion possède une polarisation -définie par la relation de dispersion exprimée sous forme matricielle -(\ref{eq:relation_disp_matr_froid}). Ainsi, on déduit des deux dernières -relations les relations existantes entre les composantes $\tilde{E}_{x}$ -et $\tilde{E}_{y}$\footnote{En convention $-j\omega t$ : $\frac{\tilde{E}_{y}}{\tilde{E}_{x}}=-\frac{jD}{S-n^{2}}$ -et $\frac{\tilde{E}_{z}}{\tilde{E}_{x}}=-\frac{n_{\perp}n_{\parallel}}{P-n_{\perp}^{2}}$.}: - -\begin{equation} -\frac{\tilde{E}_{y}}{\tilde{E}_{x}}=\frac{jD}{S+n^{2}}\label{eq:relation_entre_composantes_X_Y} -\end{equation} -et entre les composantes $\tilde{E}_{x}$ et $\tilde{E}_{z}$: - -\begin{equation} -\frac{\tilde{E}_{z}}{\tilde{E}_{x}}=\frac{n_{\perp}n_{\parallel}}{P+n_{\perp}^{2}}\label{eq:relation_entre_composantes_X_Z} -\end{equation} - -En utilisant la relation issue de la première ligne de (\ref{eq:relation_disp_matr_froid}), -on en déduit une expression générale pour le champ radial en fonction -des composantes transverses y et z : -\begin{equation} -\tilde{E}_{x}=\frac{-jD\tilde{E}_{y}+n_{\perp}n_{\parallel}\tilde{E}_{z}}{S+n_{\parallel}^{2}}\label{eq:relation_entre_composantes_X_Y_Z} -\end{equation} -En injectant cette expression dans les deux précédentes, on trouve -: - -\begin{eqnarray} -\left[\left(S+n_{\perp}^{2}+n_{\parallel}^{2}\right)\left(S+n_{\parallel}^{2}\right)-D^{2}\right]\tilde{E}_{y}-jDn_{\perp}n_{\parallel}\tilde{E}_{z} & = & 0\label{eq:relation_entre_composantes_Y_Z_1}\\ -jDn_{\perp}n_{\parallel}\tilde{E}_{y}+\left[\left(P+n_{\perp}^{2}\right)\left(S+n_{\parallel}^{2}\right)-n_{\perp}^{2}n_{\parallel}^{2}\right]\tilde{E}_{z} & = & 0\label{eq:relation_entre_composantes_Y_Z_2} -\end{eqnarray} - - -\subsubsection{Polarisation dominante des modes lents et rapides} - -En prenant \ref{eq:relation_entre_composantes_Y_Z_1} pour $n_{\perp}^{2}=n_{\perp,F}^{2}$, -on obtiens le condition de polarisation de l'onde rapide au premier -ordre : - -\begin{equation} -\tilde{E}_{z}=0\label{eq:polarisation_FastWave} -\end{equation} -En prenant \ref{eq:relation_entre_composantes_Y_Z_2} pour $n_{\perp}^{2}=n_{\perp,S}^{2}$, -on obtiens la condition de polarisation de l'onde lente au premier -ordre : - -\begin{equation} -\tilde{E}_{y}=0\label{eq:polarisation_SlowWave} -\end{equation} - - -\subsubsection{Polarisation dans le plasma} - -En prenant \ref{eq:relation_entre_composantes_X_Z} pour $n_{\perp}^{2}=n_{\perp,S}^{2}$ -dans l'hypothèse ou $D\approx0$ et $P<0$, il vient -\begin{equation} -\frac{\tilde{E}_{z}}{\tilde{E}_{x}}=\frac{\left(-\frac{P}{S}\left(S+n_{\parallel}^{2}\right)\right)^{1/2}n_{\parallel}}{P-\frac{P}{S}\left(S+n_{\parallel}^{2}\right)}=\frac{\left(S\left(S+n_{\parallel}^{2}\right)\right)^{1/2}}{\left|P\right|^{1/2}n_{\parallel}} -\end{equation} -par conséquent, à mesure que la densité locale augmente ($\left|P\right|$ -augmente, et augmente plus vite que $S$), la polarisation dominante -est radiale ($\tilde{E}_{x}$). - -\subsection{Vitesse de phase} - -\subsection{Vitesse de groupe} - -La vitesse de groupe est donnée par\cite[§7.2, §18.5]{Brambilla1998}: - -\begin{equation} -\mathbf{v}_{g}\left(\mathbf{k}_{0}\right)=\left.\frac{\partial\omega}{\partial\mathbf{k}}\right|_{\mathbf{k}=\mathbf{k}_{0}} -\end{equation} -ou, en posant Soit $\mathcal{D}$ la partie gauche de l'équation \ref{eq:cold_plasma_dispersion_relation_n}: - -\begin{equation} -\mathbf{v}_{g}\left(\mathbf{k}_{0}\right)=-\left.{\displaystyle \frac{\frac{\partial\mathcal{D}}{\partial\mathbf{k}}}{\frac{\partial\mathcal{D}}{\partial\omega}}}\right|_{\mathbf{k}=\mathbf{k}_{0}} -\end{equation} - -Dans un milieu anisotrope comme le plasma froid, la vitesse de groupe, -qui est la vitesse de propagation de l'énergie, n'est généralement -pas colinéaire avec le vecteur d'onde \textbf{$\mathbf{k}$}. Aussi, -on décompose génarelement la vitesse de groupe selon ses composantes -parallèles et perpendiculaires au vecteur d'onde. Soit -\begin{equation} -\mathbf{v}_{g}=v_{gr}\mathbf{\hat{k}}+v_{g\theta}\mathbf{\hat{e}}_{\theta} -\end{equation} -avec $\mathbf{\hat{k}}=\mathbf{k}/k_{0}$ et $\mathbf{\hat{e}}_{\theta}=\mathbf{\hat{k}}\times\left(\mathbf{B}_{0}\times\mathbf{\hat{k}}\right)/B_{0}$ -le vecteur unité perpendiculaire à $\mathbf{\hat{k}}$ dans le plan -formé par $\mathbf{B}_{0}$ et $\mathbf{\hat{k}}$, et - -\begin{equation} -v_{gr}=-\frac{1}{k_{0}}\frac{\partial\mathcal{D}/\partial n}{\partial\mathcal{D}/\partial\omega}\quad\quad v_{g\theta}=-\frac{1}{nk_{0}}\frac{\partial\mathcal{D}/\partial\theta}{\partial\mathcal{D}/\partial\omega} -\end{equation} - -Appliqué à \ref{eq:cold_plasma_dispersion_relation_n}, on a : -\begin{equation} -\end{equation} - -L'angle entre le vecteur d'onde $\mathbf{k}$ et la vitesse de groupe -est alors -\begin{equation} -\tan\alpha_{g}=\frac{v_{g\theta}}{v_{gr}}=\frac{1}{n}\frac{\partial\mathcal{D}/\partial\theta}{\partial\mathcal{D}/\partial n}=-\frac{1}{n}\frac{dn}{d\theta} -\end{equation} - - -\subsection{Réflexion/Transmission d'une onde plane par un plasma froid magnétisé} - -Soit une onde plane incidente en provenance d'un milieu de permittivité -$\varepsilon^{i}$ sur un plasma magnétisé froid. Le champ incident -a pour expression générale $\mathbf{E}^{i}e^{-j\mathbf{k}^{i}\cdot\mathbf{r}}$, -le champ réfléchis $\mathbf{E}^{r}e^{-j\mathbf{k}^{r}\cdot\mathbf{r}}$ -et le champ transmis $\mathbf{E}^{t}e^{-j\mathbf{k}^{t}\cdot\mathbf{r}}$. -Le vecteur d'onde de l'onde plane incidente $\mathbf{k}^{i}$ est -contenu dans le plan $xOz$ et forme un angle $\theta^{i}$ avec l'axe -$z$ (Figure \ref{fig:G=0000E9om=0000E9trie-reflexion}), i.e. $\mathbf{k}^{i}=\sqrt{\varepsilon^{i}}k_{0}\left(\sin\theta^{i}\mathbf{\hat{x}}+\cos\theta^{i}\mathbf{\hat{z}}\right)$. -Supposons par ailleurs que le champ électrique \textbf{$\mathbf{E}^{i}$} -soit également inclus dans ce même plan, ie. $\mathbf{E}^{i}=E^{i}\left(\cos\theta^{i}\mathbf{\hat{x}}-\sin\theta^{i}\mathbf{\hat{z}}\right)$ -(mode TM). Les conditions aux limites établissent que les composantes -transverses du champ électrique doivent être continues à l'interface, -soit\footnote{En toute généralité, on doit avoir $\mathbf{E}_{t}^{i}+\mathbf{E}_{t}^{r}=\mathbf{E}_{t}^{t}$ -à $x=0$ avec $\mathbf{E}_{t}=\mathbf{\hat{x}\times\left(\mathbf{E}\times\hat{x}\right)}$.}. -\begin{equation} -\mathbf{E}_{z}^{i}e^{-j\mathbf{k}^{i}\cdot\mathbf{r}}+\mathbf{E}_{z}^{r}e^{-j\mathbf{k}^{r}\cdot\mathbf{r}}=\mathbf{E}_{z}^{t}e^{-j\mathbf{k}^{t}\cdot\mathbf{r}} -\end{equation} -de plus pour que les deux cotés correspondent en tous points $x$ -et $y$ (\emph{phase matching}): -\begin{equation} -e^{-j\mathbf{k}^{i}\cdot\mathbf{r}}=e^{-j\mathbf{k}^{r}\cdot\mathbf{r}}=e^{-j\mathbf{k}^{t}\cdot\mathbf{r}}\quad\mbox{pour }x=0 -\end{equation} -ce qui nécessite -\begin{eqnarray} -k_{z}^{i} & =k_{z}^{r} & =k_{z}^{t}\\ -k_{y}^{i} & =k_{y}^{r} & =k_{y}^{t} -\end{eqnarray} -Puisque par hypothèse $k_{y}^{i}=0$, alors toutes les composantes -selon $y$ des vecteurs d'onde sont nulles, impliquant que les plans -d'incidente et de réflexion correspondent au plan $xOz$. D'autre -part, puisque le milieu des ondes incidentes et réfléchies est le -même, on a également $\theta^{i}=\theta^{r}$. - -\begin{figure}[h] -\begin{centering} -\includegraphics[width=0.6\textwidth]{figures/geometrie_reflexionTransmission_plasma}\caption{Géométrie du problème.\label{fig:G=0000E9om=0000E9trie-reflexion}} -\par\end{centering} -\end{figure} - -Au voisinage des fréquences LH, le milieu plasma peut s'approcher -par un milieu diélectrique uniaxe de tenseur de permittivité relative -\begin{equation} -\mathbb{K}=\left(\begin{array}{ccc} -S & 0 & 0\\ -0 & S & 0\\ -0 & 0 & P -\end{array}\right)=S\left(\mathbf{\hat{x}\mathbf{\hat{x}}+\mathbf{\hat{y}}\mathbf{\hat{y}}}\right)+P\mathbf{\,\hat{z}}\mathbf{\hat{z}} -\end{equation} - -Dans un milieu biréfringent, le vecteur de Poynting n'est pas dirigé -selon le vecteur d'onde $\mathbf{k}$ et le champ électrique n'est -pas orthogonal à $\mathbf{k}$. La simple relation de dispersion $k=n\frac{\omega}{c}$ -n'est donc plus valide. La relation de dispersion d'un tel milieu -donne, on l'a vu, les deux solutions, données par exemple par l'équation -\ref{eq:cold_plasma_dispersion_relation_solution_n}. En ce qui concerne -l'onde lente, l'indice perpendiculaire selon $x$ a pour expression -d'après la solution de l'équation de dispersion : -\begin{equation} -n_{x}^{2}=-\frac{P}{S}\left(n_{z}^{2}-S\right) -\end{equation} - -En utilisant l'égalité tirée de la condition de \emph{phase matching} -on a alors -\begin{equation} -n_{x}^{2}=-\frac{P}{S}\left(S+\varepsilon^{i}\cos^{2}\theta^{i}\right)\rightarrow n_{x}=\pm\sqrt{\frac{\left|P\right|}{S}\left(\varepsilon^{i}\cos^{2}\theta^{i}-S\right)} -\end{equation} -La direction du nombre d'onde dans le plasma et le signe de l'équation -précédente reste à déterminer. Pour choisir le signe correct, on doit -choisir le signe du flux de puissance (vecteur de Poynting) depuis -l'interface vers le plasma. D'après l'équation d'onde, on déduit -\begin{eqnarray} -n_{x}n_{z}E_{x}^{t} & = & \left(P+n_{x}^{2}\right)E_{z}^{t}\\ -\end{eqnarray} -et en utilisant les conditions de continuité du champ électrique et -de la densité flux électrique: -\begin{eqnarray} -\left(E^{r}-E^{i}\right)\sin\theta^{i} & = & E_{z}^{t}\\ -\varepsilon^{i}\left(E^{r}+E^{i}\right)\cos\theta^{i} & = & SE_{x}^{t} -\end{eqnarray} -On dispose maintenant de trois équations et trois inconnues ($E_{x}^{t},E_{z}^{t},E^{r}$) -qui nous permet de résoudre le problème: -\begin{eqnarray} -E_{x}^{t} & = & \frac{P+n_{x}^{2}}{n_{x}n_{z}}E_{z}^{t}\\ -E_{z}^{t} & = & \left(E^{r}-E^{i}\right)\sin\theta^{i}\\ -E^{r} & = & ... -\end{eqnarray} - Pour que le flux de puissance soit positif, on doit avoir -\begin{equation} -\mathbf{\hat{x}\cdot}\mathbf{S}^{t}>0\rightarrow\mathbf{\hat{x}\cdot\left(\mathbf{E}^{t}\times\mathbf{H}^{t}\right)/2} -\end{equation} -or, - -\begin{eqnarray} -\mathbf{\hat{x}\cdot}\mathbf{S}^{t} & = & \mathbf{\hat{x}\cdot\left(\mathbf{E}^{t}\times\mathbf{H}^{t}\right)/2}\\ - & = & \mathbf{H}^{t}\cdot\left(\mathbf{\hat{x}\times}\mathbf{E}^{t}\right)/2\\ - & = & \frac{1}{\omega\mu_{0}}\left(k_{z}E_{x}^{t}-k_{x}E_{z}^{t}\right)\cdot\left(-E_{z}^{t}\right)/2\mathbf{\hat{y}}\\ - & = & \frac{-k_{0}}{\omega\mu_{0}}\left(\frac{P+n_{x}^{2}}{n_{x}}-n_{x}\right)\cdot\left(E_{z}^{t}\right)^{2}/2\mathbf{\hat{y}}\\ - & = & \frac{-k_{0}}{\omega\mu_{0}}\frac{P}{n_{x}}\cdot\left(E_{z}^{t}\right)^{2}/2\mathbf{\hat{y}} -\end{eqnarray} - -Pour que la dernière expression soit positive, il est nécessaire que -$n_{x}>0$ (car $P<0$). C'est donc la racine positive qui doit être -choisie ==> Problème: cela devrait être la négative !: -\begin{equation} -n_{x}=\pm\sqrt{\frac{\left|P\right|}{S}\left(\varepsilon^{i}\cos^{2}\theta^{i}-S\right)} -\end{equation} - -Cela démontre que l'onde est backward, dans la mesure où -\end{document} +\chapter{Dispersion Relation} +\section{Vacuum dispersion relation} +In a non-dispersive isotropic homogeneous non-lossy dielectric medium, such as vacuum (or air), the wavevector $\mathbf{k}=k\mathbf{\hat{k}}$ direction is given by: +\begin{equation} + \mathbf{\hat{k}}\cdot\mathbf{E}=0 + \;\;\;\;\;\; + \mathbf{H}=\frac{n}{Z_{0}}\hat{\mathbf{k}}\times\mathbf{E}\label{eq:vecteur_onde_vide} +\end{equation} +where $\mathbf{\hat{k}}$ the unit vector in the direction of $\mathbf{k}$, $k=|\mathbf{k}|=n k_{0}$ is the wavenumber and $n=\sqrt{\varepsilon_{r}}$ the medium optical index. $Z_{0}=\sqrt{\frac{\mu_{0}}{\varepsilon_{0}}}=\mu_{0}c_{0}=\frac{1}{\varepsilon_{0}c_{0}}$ is the \emph{vacuum impedance}. The wave equation in such a medium is: +\begin{equation} + \mathbf{k} \times \mathbf{k} \times \mathbf{E} + k^{2} \mathbf{E} = \mathbf{0} +\end{equation} + +The condition for that system of 3 equations and 3 unknowns $(E_{x},E_{y},E_{z})$ to have a non-trivial solution, consists in solving the \emph{dispersion relation} between the wavevector $\mathbf{k}$ and the angular frequency $\omega$, i.e. $\mathbf{k}(\omega)$. In this medium, this relation is simple (TODO demo): $k=\sqrt{\mu\varepsilon}\omega=\frac{c}{n}\omega$\parencite[(7.4)]{Jackson1998}\footnote{While in a lossy medium $k^{2}=\mu\varepsilon\omega^{2}-j\omega\mu\sigma$ \parencite[sec. 8.2]{Bladel2007}.}. + + +% Dans un plasma froid magnétisé, la situation est radicalement différente, +% car les courants de polarisation générés par les mouvements électroniques +% et ioniques modifient la polarisation des ondes planes ainsi que leur +% dispersion. Différentes branches de dispersion, ou \emph{modes}, apparaissent\cite[chap.8]{Rax2005}. +% +% On rappelle l'expression des équations de Maxwell en régime harmonique +% pour une onde plane de vecteur d'onde $\mathbf{k}$ : +% +% \begin{eqnarray} +% \mathbf{k}\times\mathbf{E} & = & \omega\mu_{0}\mathbf{H}\\ +% \mathbf{k}\times\mathbf{H} & = & -\omega\varepsilon_{0}\mathbf{K}\cdot\mathbf{E} +% \end{eqnarray} +% Pour déterminer les propriétés de ces modes, on étudie les solutions +% de l'\emph{équation d'onde} déduite des deux précédentes équations\footnote{NB : L'équation d'onde ne dépend pas de la convention temporelle choisie.} +% : +% +% \begin{equation} +% \mathbf{n}\times\mathbf{n}\times\mathbf{\tilde{E}}+\mathbb{K}\cdot\mathbf{\tilde{E}}=\mathbf{0}\label{eq:Helmoltz} +% \end{equation} +% où $\mathbf{n}=\mathbf{k}/k_{0}$ correspond au vecteur d'indice de +% réfraction, dont la direction est celle du vecteur d'onde $\mathbf{k}$ +% et l'amplitude celle de l'indice de réfraction. A priori, la matrice +% $\mathbb{K}$ dépend des trois composantes du vecteur d'onde $\mathbf{k}$. +% En choisissant le système de coordonnées cartésien l'équation \ref{eq:Helmoltz} +% peut s'écrire sous forme matricielle\footnote{On peut s'économiser un peu de calcul vectoriel grâce à un peu d'algèbre +% et le formalisme des dyadiques\cite{Belov2003,Lindell1995}. En utilisant +% l'identité ``BAC-CAB'', le double produit vectoriel s'écrit $\mathbf{n}\times\mathbf{n}\times\mathbf{\tilde{E}}=\mathbf{n}\left(\mathbf{n}\cdot\mathbf{\tilde{E}}\right)-\mathbf{\tilde{E}}\left(\mathbf{n}\cdot\mathbf{n}\right)=\mathbf{n}\left(\mathbf{n}\cdot\mathbf{\tilde{E}}\right)-n^{2}\mathbf{\tilde{E}}$. +% Avec l'opération dyadique $\mathbf{n}\left(\mathbf{n}\cdot\tilde{\mathbf{E}}\right)=\mathbf{nn}\cdot\mathbf{\tilde{E}}$ +% et $\mathbb{I}=\mathbf{xx}+\mathbf{yy}=\mathbf{zz}$ l'opérateur dyadique +% unité vérifiant $\mathbf{\tilde{E}}=\mathbb{I}\cdot\tilde{\mathbf{E}}$on +% peut alors factoriser l'équation \ref{eq:Helmoltz} en un opérateur +% dyadique $\left(\mathbf{nn}-n^{2}\mathbb{I}+\mathbb{K}\right)\cdot\tilde{\mathbf{E}}=\mathbf{0}$ +% qui donne directement l'expression matricielle de l'équation \ref{eq:relation_disp_matr}.} : +% \begin{equation} +% \left(\begin{array}{ccc} +% K_{xx}-n_{y}^{2}-n_{z}^{2} & K_{xy}+n_{x}n_{y} & K_{xz}+n_{x}n_{z}\\ +% K_{yx}+n_{x}n_{y} & K_{yy}-n_{x}^{2}-n_{z}^{2} & K_{yz}+n_{y}n_{z}\\ +% K_{zx}+n_{x}n_{z} & K_{zy}+n_{y}n_{z} & K_{zz}-n_{x}^{2}-n_{y}^{2} +% \end{array}\right)\left(\begin{array}{c} +% \tilde{E_{x}}\\ +% \tilde{E}_{y}\\ +% \tilde{E}_{z} +% \end{array}\right)=\mathbf{0}\label{eq:relation_disp_matr} +% \end{equation} +% Si l'on suppose en plus que le vecteur d'onde $\mathbf{k}$ (ie. +% la propagation de l'onde) soit contenu dans le plan $x-z$ (ie $k_{y}=n_{y}=0$), +% l'équation précédente devient : +% +% \begin{equation} +% \left(\begin{array}{ccc} +% K_{xx}-n_{z}^{2} & K_{xy} & K_{xz}+n_{x}n_{z}\\ +% K_{yx} & K_{yy}-n_{x}^{2}-n_{z}^{2} & K_{yz}\\ +% K_{zx}+n_{x}n_{z} & K_{zy} & K_{zz}-n_{x}^{2} +% \end{array}\right)\left(\begin{array}{c} +% \tilde{E_{x}}\\ +% \tilde{E}_{y}\\ +% \tilde{E}_{z} +% \end{array}\right)=\mathbf{0}\label{eq:relation_disp_matr_full} +% \end{equation} +% Si le plasma est homogène (indépendant de $\mathbf{r}$) on peut exploiter +% l'équivalence entre toutes les directions perpendiculaires au champ +% magnétique statique pour prédire que $\mathbb{K}$ doit être fonction +% de $k_{\parallel}$et $k_{\perp}^{2}$ seulement. +% +% \begin{figure} +% \begin{centering} +% \includegraphics[width=0.5\textwidth]{figures/geometrie_dispersion_plasma} +% \par\end{centering} +% \caption{Géométrie cartésienne du milieu plasma.\label{fig:G=0000E9om=0000E9trie-cart=0000E9sienne-plasma}} +% \end{figure} +% +% Si $\mathbb{K}$ est le tenseur de permittivité d'un plasma froid +% (\ref{eq:tenseur_stix}) définit dans l'annexe \ref{sec:Tenseur-de-permittivit=0000E9}, +% c'est-à-dire tel que $z$ soit parallèle au champ magnétique (Figure +% \ref{fig:G=0000E9om=0000E9trie-cart=0000E9sienne-plasma}), alors +% on a: +% +% \begin{equation} +% \left(\begin{array}{ccc} +% S-n_{z}^{2} & jD & n_{x}n_{z}\\ +% -jD & S-n_{x}^{2}-n_{z}^{2} & 0\\ +% n_{x}n_{z} & 0 & P-n_{x}^{2} +% \end{array}\right)\left(\begin{array}{c} +% \tilde{E_{x}}\\ +% \tilde{E}_{y}\\ +% \tilde{E}_{z} +% \end{array}\right)=\mathbf{0}\label{eq:relation_disp_matr_froid} +% \end{equation} +% On définit $\theta$ comme l'angle entre le vecteur d'onde $\mathbf{k}$ +% et la direction $\hat{\mathbf{e}}_{z}$ du champ magnétique, soit +% $n_{x}=n_{\perp}=n\sin\theta$ et $n_{z}=n_{\parallel}=n\cos\theta$, +% on a alors \footnote{Pour obtenir la version harmonique en convention $-j\omega t$, il +% faut remplacer $n^{2}\rightarrow-n^{2}$ et $j\rightarrow-j$.}: +% +% \begin{equation} +% \left(\begin{array}{ccc} +% S-n^{2}\cos^{2}\theta & jD & n^{2}\cos\theta\sin\theta\\ +% -jD & S-n^{2} & 0\\ +% n^{2}\cos\theta\sin\theta & 0 & P-n^{2}\sin^{2}\theta +% \end{array}\right)\left(\begin{array}{c} +% \tilde{E}_{x}\\ +% \tilde{E}_{y}\\ +% \tilde{E}_{z} +% \end{array}\right)=\mathbf{0}\label{eq:cold_plasma_dispersion_relation_matrix} +% \end{equation} +% +% L'existence de solutions non triviales à l'équation d'onde (les modes) +% (\ref{eq:relation_disp_matr_full}) nécessite que le déterminant de +% la matrice soit nul. Cette condition donne la \emph{relation de dispersion}, +% qui pour un plasma froid peut s'écrire \cite[p.8-9]{Stix1992}\cite[§2.1.3]{Swanson2003}\cite[§18.1]{Brambilla1998}: +% +% \begin{equation} +% \boxed{An^{4}-Bn^{2}+C=0}\label{eq:cold_plasma_dispersion_relation_n} +% \end{equation} +% avec\footnote{Où l'on a remarqué que $S^{2}-D^{2}=RL$. Les expressions $A,B,C$ +% ne dépendent pas de la convention temporelle choisie. }: +% +% \begin{eqnarray} +% A & = & S\sin^{2}\theta+P\cos^{2}\theta\\ +% B & = & \left(S^{2}-D^{2}\right)\sin^{2}\theta+PS\left(1+\cos^{2}\theta\right)=RL\sin^{2}\theta+PS\left(1+\cos^{2}\theta\right)\\ +% C & = & P\left(S^{2}-D^{2}\right)=PRL +% \end{eqnarray} +% Soit $n=n(\theta,\omega)=n(\hat{\mathbf{k}},\omega)$ la solution +% de la relation de dispersion pour une fréquence $\omega$ et une direction +% de propagation $\hat{\mathbf{k}}$ (ie. $\theta$) données. Une onde +% plane d'indice $n$ et de nombre d'onde $\mathbf{k}=n\frac{\omega}{c}\mathbf{\hat{k}}$ +% peut se propager dans le plasma à la fréquence $\omega$ et dans la +% direction du vecteur unitaire $\mathbf{\hat{k}}$ en l'absence de +% sources extérieures (plus exactement avec des sources situées à l'infini). +% Une telle onde est appelée \emph{onde caractéristique} ou \emph{mode +% de propagation} (mode propre) du plasma. +% +% L'équation (\ref{eq:cold_plasma_dispersion_relation_n}) est une équation +% du second degré en $n^{2}$ ayant pour solution : +% \begin{equation} +% n^{2}=\frac{B\pm\sqrt{\Delta}}{2A}\label{eq:solution_cold_plasma_dispersion_relation_n} +% \end{equation} +% où son déterminant $\Delta=B^{2}-4AC$ vaut : +% \begin{equation} +% \Delta=(RL-PS)^{2}\sin^{4}\theta+4P^{2}D^{2}\cos^{2}\theta\label{eq:cold_plasma_determinant_dispersion_relation_n} +% \end{equation} +% +% Le déterminant (\ref{eq:cold_plasma_determinant_dispersion_relation_n}) +% n'est jamais négatif, ce qui signifie que l'équation (\ref{eq:cold_plasma_dispersion_relation_n}) +% a toujours deux solutions réelles et distinctes en $n^{2}$. Ainsi, +% les ondes planes dans un plasma froid sont soit purement propagatives +% ($n^{2}>0$) soit purement évanescentes ($n^{2}<0$) ; les oscillations +% amorties sont exclues. La transition entre ces deux régimes a lieu +% aux coupures ($n=0$) et aux résonances ($n\to\infty$). +% +% Les deux racines peuvent être confondues lorsque le déterminant être +% nul, dans les cas particulier suivant : +% \begin{itemize} +% \item En propagation parallèle, ie $\theta=0$, lorsque $P=0$ ; +% \item En propagation perpendiculaire, ie $\theta=\pi/2$, lorsque $RL=PS$. +% \end{itemize} +% Dans le cadre de l'approximation froide définie par l\textquoteright absence +% de \emph{dispersion spatiale}, c'est-à-dire lorsque les éléments du +% tenseur $\mathbb{K}$ ne dépendent pas de l'indice de réfraction $\mathbf{n}$ +% (ie. du vecteur d'onde $\mathbf{k}$), l'équation (\ref{eq:cold_plasma_dispersion_relation_n}) +% est une simple équation quadratique en $n^{2}$. Il existe donc deux +% solutions distinctes qui peuvent se propager dans le plasma ; un plasma +% froid est donc un milieu \emph{biréfringent} (les ondes peuvent être +% évanescentes ou non, selon les caractéristiques du plasma). Si on +% avait introduit des effets thermiques, les éléments du tenseur de +% permittivité dépendraient alors du vecteur d'onde et de nouveaux modes +% apparaitraient\cite[§1.2.2]{Dumont2007}. +% +% \subsubsection{Expression en fonction de $\theta$.} +% +% L'équation de dispersion (\ref{eq:cold_plasma_dispersion_relation_n}) +% peut être exprimée sous diverses formes équivalentes. La relation +% de dispersion (\ref{eq:cold_plasma_dispersion_relation_n}) peut être +% exprimée en fonction de l'angle $\theta$\cite[§18.2]{Brambilla1998}: +% +% \begin{equation} +% \tan^{2}\theta=-\frac{P\left(n^{2}-R\right)\left(n^{2}-L\right)}{\left(Sn^{2}-RL\right)\left(n^{2}-P\right)}\label{eq:cold_plasma_dispersion_relation_theta} +% \end{equation} +% En résolvant pour $n^{2}$ on obtient un cas particulier des équations +% d'Appleton-Hartree, et en particulier pour $D=0$ +% \begin{equation} +% n^{2}=\begin{cases} +% \frac{PS}{S\sin^{2}\theta+P\cos^{2}\theta} & \mbox{extraordinary mode}\\ +% S & \mbox{oridnary mode} +% \end{cases}\label{eq:cold_plasma_dispersion_relation_solution_n} +% \end{equation} +% +% +% \subsubsection{Expression en fonction de $n_{\parallel}$.} +% +% Lorsque le nombre d'onde $n_{\parallel}=n_{z}$ est définit par des +% conditions extérieures, comme la structure d'une antenne, et en supposant +% que la propagation est contrainte au plan (xOz) ($n_{y}=0$), la relation +% de dispersion peut être exprimée en fonction de $n_{\perp}^{2}=n_{x}^{2}=n^{2}-n_{\parallel}^{2}$. +% Ainsi, le déterminant de (\ref{eq:relation_disp_matr_froid}) donne +% une équations quadratique en $n_{\perp}^{2}$\cite[§18.2]{Brambilla1998}\footnote{En convention $-j\omega t$: $A_{1}n_{\perp}^{4}-B_{1}n_{\perp}^{2}+C_{1}=0$ }: +% +% \begin{equation} +% A_{1}n_{\perp}^{4}+B_{1}n_{\perp}^{2}+C_{1}=0\label{eq:cold_plasma_dispersion_relation_n_perp} +% \end{equation} +% avec +% +% \begin{eqnarray} +% A_{1} & = & S\\ +% B_{1} & = & \left(P+S\right)\left(n_{\parallel}^{2}-S\right)+D^{2}=RL+PS-n_{\parallel}^{2}(P+S)\\ +% C_{1} & = & P\left(\left(n_{\parallel}^{2}-S\right)^{2}-D^{2}\right)=P(n_{\parallel}^{2}-R)(n_{\parallel}^{2}-L) +% \end{eqnarray} +% où on rappelle que $R=S+D$ et $L=S-D$. Les coupures, qui correspondent +% aux transitions entre propagation et évanescence, sont alors définies +% pour $n_{\perp}=0$. Les résonances pour $n_{\perp}\to\infty$. Comme +% précédemment, le déterminant de l'équation est toujours positif ou +% nul, l'équation (\ref{eq:cold_plasma_dispersion_relation_n_perp}) +% possède donc deux solutions, qui peuvent éventuellement être complexes +% conjuguées selon les conditions (fréquence, densité, etc.). La solution +% s'exprime par \cite[§15.9.2]{Friedberg2007}: +% +% \begin{equation} +% n_{\perp}^{2}=-\frac{P}{2S}\left(D^{2}/P-S-n_{\parallel}^{2}\pm\sqrt{\left(D^{2}/P-S-n_{\parallel}^{2}\right)^{2}+\frac{4SD^{2}}{P}}\right)\label{eq:cold_plasma_solution_n_perp} +% \end{equation} +% La solution dont la valeur est la plus grande, c'est-à-dire pour laquelle +% la valeur de la vitesse de phase perpendiculaire $v_{\perp}=\omega/k_{\perp}$ +% sera la plus petite, correspond à la branche dite \emph{lente}, l'autre +% branche étant la solution dite \emph{rapide}. +% +% \begin{figure}[h] +% \begin{centering} +% \includegraphics[width=0.9\textwidth]{figures/n_perp_vs_ne}\label{Flo:n_perp_vs_ne}\caption{Exemple de solutions de l'équation de dispersion pour $n_{\perp}^{2}$ +% en fonction de la densité électronique $n_{e}$. \textbf{$n_{\parallel}=2.0$ +% }et\textbf{ $B_{0}=2.95$~}T, f=3.7GHz.} +% \par\end{centering} +% \end{figure} +% +% Pour des valeurs de $\left|P\right|$ grande ($\omega\ll\omega_{pe}$), +% les deux racines (\ref{eq:cold_plasma_solution_n_perp}) peuvent s'approcher +% par au premier ordre en $\omega^{2}/\omega_{pe}^{2}$ (cf. \cite[p.222]{Brambilla1998}). +% De la même façon, dans le voisinage des fréquences LH, on peut faire +% l'hypothèse $D\approx0$. Dans ce cas, l'équation de dispersion s'écrit +% (avec la convention temporelle $e^{j\omega t}$): +% \begin{eqnarray} +% A_{1}n_{\perp}^{4}+B_{1}n_{\perp}^{2}+C_{1} & = & 0\label{eq:cold_plasma_dispersion_relation_n_perp_approx1} +% \end{eqnarray} +% avec +% \begin{eqnarray} +% A_{1} & = & S\\ +% B_{1} & = & S^{2}+PS-n_{\parallel}^{2}(P+S)\\ +% C_{1} & = & P(n_{\parallel}^{2}-S)^{2} +% \end{eqnarray} +% Les solutions de l'équation de dispersion \ref{eq:cold_plasma_dispersion_relation_n_perp_approx1} +% sont dans ce cas : +% \begin{eqnarray} +% n_{\perp}^{2}=n_{\perp,F}^{2} & = & -\left(S+n_{\parallel}^{2}\right)\label{eq:n_perp_fast_wave_solution_approximate}\\ +% n_{\perp}^{2}=n_{\perp,S}^{2} & = & -\frac{P}{S}\left(S+n_{\parallel}^{2}\right)\label{eq:n_perp_slow_wave_solution_approximate} +% \end{eqnarray} +% Enfin, toujours au voisinage du domaine de plasma de bord, $S\approx1$ +% d'où \ref{eq:cold_plasma_dispersion_relation_n_perp_approx1} : +% \begin{equation} +% n_{\perp}^{4}+\left(1-n_{\parallel}^{2}\right)\left(1+P\right)n_{\perp}^{2}+P\left(n_{\parallel}^{2}-1\right)^{2}=0 +% \end{equation} +% qui a pour solutions: +% \begin{eqnarray} +% n_{\perp}^{2}=n_{\perp,F}^{2} & = & -\left(1-n_{\parallel}^{2}\right)\label{eq:n_perp_fast_wave_solution_approximate2}\\ +% n_{\perp}^{2}=n_{\perp,S}^{2} & = & -P\left(1-n_{\parallel}^{2}\right)\label{eq:n_perp_slow_wave_solution_approximate2} +% \end{eqnarray} +% +% +% \subsection{Polarisation des champs} +% +% Chaque mode propre de la solution de dispersion possède une polarisation +% définie par la relation de dispersion exprimée sous forme matricielle +% (\ref{eq:relation_disp_matr_froid}). Ainsi, on déduit des deux dernières +% relations les relations existantes entre les composantes $\tilde{E}_{x}$ +% et $\tilde{E}_{y}$\footnote{En convention $-j\omega t$ : $\frac{\tilde{E}_{y}}{\tilde{E}_{x}}=-\frac{jD}{S-n^{2}}$ +% et $\frac{\tilde{E}_{z}}{\tilde{E}_{x}}=-\frac{n_{\perp}n_{\parallel}}{P-n_{\perp}^{2}}$.}: +% +% \begin{equation} +% \frac{\tilde{E}_{y}}{\tilde{E}_{x}}=\frac{jD}{S+n^{2}}\label{eq:relation_entre_composantes_X_Y} +% \end{equation} +% et entre les composantes $\tilde{E}_{x}$ et $\tilde{E}_{z}$: +% +% \begin{equation} +% \frac{\tilde{E}_{z}}{\tilde{E}_{x}}=\frac{n_{\perp}n_{\parallel}}{P+n_{\perp}^{2}}\label{eq:relation_entre_composantes_X_Z} +% \end{equation} +% +% En utilisant la relation issue de la première ligne de (\ref{eq:relation_disp_matr_froid}), +% on en déduit une expression générale pour le champ radial en fonction +% des composantes transverses y et z : +% \begin{equation} +% \tilde{E}_{x}=\frac{-jD\tilde{E}_{y}+n_{\perp}n_{\parallel}\tilde{E}_{z}}{S+n_{\parallel}^{2}}\label{eq:relation_entre_composantes_X_Y_Z} +% \end{equation} +% En injectant cette expression dans les deux précédentes, on trouve +% : +% +% \begin{eqnarray} +% \left[\left(S+n_{\perp}^{2}+n_{\parallel}^{2}\right)\left(S+n_{\parallel}^{2}\right)-D^{2}\right]\tilde{E}_{y}-jDn_{\perp}n_{\parallel}\tilde{E}_{z} & = & 0\label{eq:relation_entre_composantes_Y_Z_1}\\ +% jDn_{\perp}n_{\parallel}\tilde{E}_{y}+\left[\left(P+n_{\perp}^{2}\right)\left(S+n_{\parallel}^{2}\right)-n_{\perp}^{2}n_{\parallel}^{2}\right]\tilde{E}_{z} & = & 0\label{eq:relation_entre_composantes_Y_Z_2} +% \end{eqnarray} +% +% +% \subsubsection{Polarisation dominante des modes lents et rapides} +% +% En prenant \ref{eq:relation_entre_composantes_Y_Z_1} pour $n_{\perp}^{2}=n_{\perp,F}^{2}$, +% on obtiens le condition de polarisation de l'onde rapide au premier +% ordre : +% +% \begin{equation} +% \tilde{E}_{z}=0\label{eq:polarisation_FastWave} +% \end{equation} +% En prenant \ref{eq:relation_entre_composantes_Y_Z_2} pour $n_{\perp}^{2}=n_{\perp,S}^{2}$, +% on obtiens la condition de polarisation de l'onde lente au premier +% ordre : +% +% \begin{equation} +% \tilde{E}_{y}=0\label{eq:polarisation_SlowWave} +% \end{equation} +% +% +% \subsubsection{Polarisation dans le plasma} +% +% En prenant \ref{eq:relation_entre_composantes_X_Z} pour $n_{\perp}^{2}=n_{\perp,S}^{2}$ +% dans l'hypothèse ou $D\approx0$ et $P<0$, il vient +% \begin{equation} +% \frac{\tilde{E}_{z}}{\tilde{E}_{x}}=\frac{\left(-\frac{P}{S}\left(S+n_{\parallel}^{2}\right)\right)^{1/2}n_{\parallel}}{P-\frac{P}{S}\left(S+n_{\parallel}^{2}\right)}=\frac{\left(S\left(S+n_{\parallel}^{2}\right)\right)^{1/2}}{\left|P\right|^{1/2}n_{\parallel}} +% \end{equation} +% par conséquent, à mesure que la densité locale augmente ($\left|P\right|$ +% augmente, et augmente plus vite que $S$), la polarisation dominante +% est radiale ($\tilde{E}_{x}$). +% +% \subsection{Vitesse de phase} +% +% \subsection{Vitesse de groupe} +% +% La vitesse de groupe est donnée par\cite[§7.2, §18.5]{Brambilla1998}: +% +% \begin{equation} +% \mathbf{v}_{g}\left(\mathbf{k}_{0}\right)=\left.\frac{\partial\omega}{\partial\mathbf{k}}\right|_{\mathbf{k}=\mathbf{k}_{0}} +% \end{equation} +% ou, en posant Soit $\mathcal{D}$ la partie gauche de l'équation \ref{eq:cold_plasma_dispersion_relation_n}: +% +% \begin{equation} +% \mathbf{v}_{g}\left(\mathbf{k}_{0}\right)=-\left.{\displaystyle \frac{\frac{\partial\mathcal{D}}{\partial\mathbf{k}}}{\frac{\partial\mathcal{D}}{\partial\omega}}}\right|_{\mathbf{k}=\mathbf{k}_{0}} +% \end{equation} +% +% Dans un milieu anisotrope comme le plasma froid, la vitesse de groupe, +% qui est la vitesse de propagation de l'énergie, n'est généralement +% pas colinéaire avec le vecteur d'onde \textbf{$\mathbf{k}$}. Aussi, +% on décompose génarelement la vitesse de groupe selon ses composantes +% parallèles et perpendiculaires au vecteur d'onde. Soit +% \begin{equation} +% \mathbf{v}_{g}=v_{gr}\mathbf{\hat{k}}+v_{g\theta}\mathbf{\hat{e}}_{\theta} +% \end{equation} +% avec $\mathbf{\hat{k}}=\mathbf{k}/k_{0}$ et $\mathbf{\hat{e}}_{\theta}=\mathbf{\hat{k}}\times\left(\mathbf{B}_{0}\times\mathbf{\hat{k}}\right)/B_{0}$ +% le vecteur unité perpendiculaire à $\mathbf{\hat{k}}$ dans le plan +% formé par $\mathbf{B}_{0}$ et $\mathbf{\hat{k}}$, et +% +% \begin{equation} +% v_{gr}=-\frac{1}{k_{0}}\frac{\partial\mathcal{D}/\partial n}{\partial\mathcal{D}/\partial\omega}\quad\quad v_{g\theta}=-\frac{1}{nk_{0}}\frac{\partial\mathcal{D}/\partial\theta}{\partial\mathcal{D}/\partial\omega} +% \end{equation} +% +% Appliqué à \ref{eq:cold_plasma_dispersion_relation_n}, on a : +% \begin{equation} +% \end{equation} +% +% L'angle entre le vecteur d'onde $\mathbf{k}$ et la vitesse de groupe +% est alors +% \begin{equation} +% \tan\alpha_{g}=\frac{v_{g\theta}}{v_{gr}}=\frac{1}{n}\frac{\partial\mathcal{D}/\partial\theta}{\partial\mathcal{D}/\partial n}=-\frac{1}{n}\frac{dn}{d\theta} +% \end{equation} +% +% +% \subsection{Réflexion/Transmission d'une onde plane par un plasma froid magnétisé} +% +% Soit une onde plane incidente en provenance d'un milieu de permittivité +% $\varepsilon^{i}$ sur un plasma magnétisé froid. Le champ incident +% a pour expression générale $\mathbf{E}^{i}e^{-j\mathbf{k}^{i}\cdot\mathbf{r}}$, +% le champ réfléchis $\mathbf{E}^{r}e^{-j\mathbf{k}^{r}\cdot\mathbf{r}}$ +% et le champ transmis $\mathbf{E}^{t}e^{-j\mathbf{k}^{t}\cdot\mathbf{r}}$. +% Le vecteur d'onde de l'onde plane incidente $\mathbf{k}^{i}$ est +% contenu dans le plan $xOz$ et forme un angle $\theta^{i}$ avec l'axe +% $z$ (Figure \ref{fig:G=0000E9om=0000E9trie-reflexion}), i.e. $\mathbf{k}^{i}=\sqrt{\varepsilon^{i}}k_{0}\left(\sin\theta^{i}\mathbf{\hat{x}}+\cos\theta^{i}\mathbf{\hat{z}}\right)$. +% Supposons par ailleurs que le champ électrique \textbf{$\mathbf{E}^{i}$} +% soit également inclus dans ce même plan, ie. $\mathbf{E}^{i}=E^{i}\left(\cos\theta^{i}\mathbf{\hat{x}}-\sin\theta^{i}\mathbf{\hat{z}}\right)$ +% (mode TM). Les conditions aux limites établissent que les composantes +% transverses du champ électrique doivent être continues à l'interface, +% soit\footnote{En toute généralité, on doit avoir $\mathbf{E}_{t}^{i}+\mathbf{E}_{t}^{r}=\mathbf{E}_{t}^{t}$ +% à $x=0$ avec $\mathbf{E}_{t}=\mathbf{\hat{x}\times\left(\mathbf{E}\times\hat{x}\right)}$.}. +% \begin{equation} +% \mathbf{E}_{z}^{i}e^{-j\mathbf{k}^{i}\cdot\mathbf{r}}+\mathbf{E}_{z}^{r}e^{-j\mathbf{k}^{r}\cdot\mathbf{r}}=\mathbf{E}_{z}^{t}e^{-j\mathbf{k}^{t}\cdot\mathbf{r}} +% \end{equation} +% de plus pour que les deux cotés correspondent en tous points $x$ +% et $y$ (\emph{phase matching}): +% \begin{equation} +% e^{-j\mathbf{k}^{i}\cdot\mathbf{r}}=e^{-j\mathbf{k}^{r}\cdot\mathbf{r}}=e^{-j\mathbf{k}^{t}\cdot\mathbf{r}}\quad\mbox{pour }x=0 +% \end{equation} +% ce qui nécessite +% \begin{eqnarray} +% k_{z}^{i} & =k_{z}^{r} & =k_{z}^{t}\\ +% k_{y}^{i} & =k_{y}^{r} & =k_{y}^{t} +% \end{eqnarray} +% Puisque par hypothèse $k_{y}^{i}=0$, alors toutes les composantes +% selon $y$ des vecteurs d'onde sont nulles, impliquant que les plans +% d'incidente et de réflexion correspondent au plan $xOz$. D'autre +% part, puisque le milieu des ondes incidentes et réfléchies est le +% même, on a également $\theta^{i}=\theta^{r}$. +% +% \begin{figure}[h] +% \begin{centering} +% \includegraphics[width=0.6\textwidth]{figures/geometrie_reflexionTransmission_plasma}\caption{Géométrie du problème.\label{fig:G=0000E9om=0000E9trie-reflexion}} +% \par\end{centering} +% \end{figure} +% +% Au voisinage des fréquences LH, le milieu plasma peut s'approcher +% par un milieu diélectrique uniaxe de tenseur de permittivité relative +% \begin{equation} +% \mathbb{K}=\left(\begin{array}{ccc} +% S & 0 & 0\\ +% 0 & S & 0\\ +% 0 & 0 & P +% \end{array}\right)=S\left(\mathbf{\hat{x}\mathbf{\hat{x}}+\mathbf{\hat{y}}\mathbf{\hat{y}}}\right)+P\mathbf{\,\hat{z}}\mathbf{\hat{z}} +% \end{equation} +% +% Dans un milieu biréfringent, le vecteur de Poynting n'est pas dirigé +% selon le vecteur d'onde $\mathbf{k}$ et le champ électrique n'est +% pas orthogonal à $\mathbf{k}$. La simple relation de dispersion $k=n\frac{\omega}{c}$ +% n'est donc plus valide. La relation de dispersion d'un tel milieu +% donne, on l'a vu, les deux solutions, données par exemple par l'équation +% \ref{eq:cold_plasma_dispersion_relation_solution_n}. En ce qui concerne +% l'onde lente, l'indice perpendiculaire selon $x$ a pour expression +% d'après la solution de l'équation de dispersion : +% \begin{equation} +% n_{x}^{2}=-\frac{P}{S}\left(n_{z}^{2}-S\right) +% \end{equation} +% +% En utilisant l'égalité tirée de la condition de \emph{phase matching} +% on a alors +% \begin{equation} +% n_{x}^{2}=-\frac{P}{S}\left(S+\varepsilon^{i}\cos^{2}\theta^{i}\right)\rightarrow n_{x}=\pm\sqrt{\frac{\left|P\right|}{S}\left(\varepsilon^{i}\cos^{2}\theta^{i}-S\right)} +% \end{equation} +% La direction du nombre d'onde dans le plasma et le signe de l'équation +% précédente reste à déterminer. Pour choisir le signe correct, on doit +% choisir le signe du flux de puissance (vecteur de Poynting) depuis +% l'interface vers le plasma. D'après l'équation d'onde, on déduit +% \begin{eqnarray} +% n_{x}n_{z}E_{x}^{t} & = & \left(P+n_{x}^{2}\right)E_{z}^{t}\\ +% \end{eqnarray} +% et en utilisant les conditions de continuité du champ électrique et +% de la densité flux électrique: +% \begin{eqnarray} +% \left(E^{r}-E^{i}\right)\sin\theta^{i} & = & E_{z}^{t}\\ +% \varepsilon^{i}\left(E^{r}+E^{i}\right)\cos\theta^{i} & = & SE_{x}^{t} +% \end{eqnarray} +% On dispose maintenant de trois équations et trois inconnues ($E_{x}^{t},E_{z}^{t},E^{r}$) +% qui nous permet de résoudre le problème: +% \begin{eqnarray} +% E_{x}^{t} & = & \frac{P+n_{x}^{2}}{n_{x}n_{z}}E_{z}^{t}\\ +% E_{z}^{t} & = & \left(E^{r}-E^{i}\right)\sin\theta^{i}\\ +% E^{r} & = & ... +% \end{eqnarray} +% Pour que le flux de puissance soit positif, on doit avoir +% \begin{equation} +% \mathbf{\hat{x}\cdot}\mathbf{S}^{t}>0\rightarrow\mathbf{\hat{x}\cdot\left(\mathbf{E}^{t}\times\mathbf{H}^{t}\right)/2} +% \end{equation} +% or, +% +% \begin{eqnarray} +% \mathbf{\hat{x}\cdot}\mathbf{S}^{t} & = & \mathbf{\hat{x}\cdot\left(\mathbf{E}^{t}\times\mathbf{H}^{t}\right)/2}\\ +% & = & \mathbf{H}^{t}\cdot\left(\mathbf{\hat{x}\times}\mathbf{E}^{t}\right)/2\\ +% & = & \frac{1}{\omega\mu_{0}}\left(k_{z}E_{x}^{t}-k_{x}E_{z}^{t}\right)\cdot\left(-E_{z}^{t}\right)/2\mathbf{\hat{y}}\\ +% & = & \frac{-k_{0}}{\omega\mu_{0}}\left(\frac{P+n_{x}^{2}}{n_{x}}-n_{x}\right)\cdot\left(E_{z}^{t}\right)^{2}/2\mathbf{\hat{y}}\\ +% & = & \frac{-k_{0}}{\omega\mu_{0}}\frac{P}{n_{x}}\cdot\left(E_{z}^{t}\right)^{2}/2\mathbf{\hat{y}} +% \end{eqnarray} +% +% Pour que la dernière expression soit positive, il est nécessaire que +% $n_{x}>0$ (car $P<0$). C'est donc la racine positive qui doit être +% choisie ==> Problème: cela devrait être la négative !: +% \begin{equation} +% n_{x}=\pm\sqrt{\frac{\left|P\right|}{S}\left(\varepsilon^{i}\cos^{2}\theta^{i}-S\right)} +% \end{equation} +% +% Cela démontre que l'onde est backward, dans la mesure où diff --git a/HCD_Textbook.bib b/HCD_Textbook.bib index c0b31e7..050c797 100644 --- a/HCD_Textbook.bib +++ b/HCD_Textbook.bib @@ -1,119 +1,20 @@ -Automatically generated by Mendeley Desktop 1.17.10 +Automatically generated by Mendeley Desktop 1.17.8 Any changes to this file will be lost if it is regenerated by Mendeley. BibTeX export options can be customized via Options -> BibTeX in Mendeley Desktop -@misc{Michelsen2017, -author = {Michelsen, Eric}, -title = {{Funky Electromagnetic Concepts}}, -url = {physics.ucsd.edu/{~}emichels}, -year = {2017} -} -@article{Peysson2012, -abstract = {A new ray-tracing code named C3P O has been developed to study the propagation of arbitrary electromagnetic radio-frequency (rf) waves in magnetized toroidal plasmas. Its structure is designed for maximum flexibility regarding the choice of coordinate system and dielectric model. The versatility of this code makes it particularly suitable for integrated modeling systems. Using a coordinate system that reflects the nested structure of magnetic flux surfaces in tokamaks, fast and accurate calculations inside the plasma separatrix can be performed using analytical derivatives of a spline-Fourier interpolation of the axisymmetric toroidal MHD equilibrium. Applications to reverse field pinch magnetic configuration are also included. The effects of 3D perturbations of the axisymmetric toroidal MHD equilibrium, due to the discreteness of the magnetic coil system or plasma fluctuations in an original quasi-optical approach, are also studied. Using a Runge–Kutta–Fehlberg method for solving the set of ordinary differential equations, the ray-tracing code is extensively benchmarked against analytical models and other codes for lower hybrid and electron cyclotron waves. (Some figures may appear in colour only in the online journal)}, -author = {Peysson, Yves and Decker, Joan and Morini, L}, -doi = {10.1088/0741-3335/54/4/045003}, -file = {::;::}, -issn = {0741-3335}, -journal = {Plasma Phys. Control. Fusion}, -number = {54}, -pages = {45003--45003}, -title = {{A versatile ray-tracing code for studying rf wave propagation in toroidal magnetized plasmas}}, -url = {http://iopscience.iop.org/0741-3335/54/4/045003}, -volume = {54}, -year = {2012} -} -@techreport{Theilhaber1979, -author = {Theilhaber, K and Bers, A}, -doi = {10.1088/0029-5515/20/5/003}, -file = {::}, -institution = {Plasma Fusion Center, MIT / JA-79-15}, -isbn = {0029-5515}, -issn = {17414326}, -title = {{Coupling to the Fast Wave at Lower Hybrid Frequencies}}, -year = {1979} -} -@article{Gormezano1986, -abstract = {The author reviews the interaction of lower hybrid waves and plasmas which is a very versatile method. The method has proven to be effective in a large range of applications which the author discusses: (1) bulk ion heating; (2) bulk electron heating; and (3) noninductive current drive.}, -author = {Gormezano, C.}, -doi = {10.1088/0741-3335/28/9A/014}, -file = {::}, -issn = {0741-3335}, -journal = {Plasma Physics and Controlled Fusion}, -month = {sep}, -number = {9A}, -pages = {1365--1376}, -title = {{Review of lower hybrid wave heating and current drive}}, -url = {http://stacks.iop.org/0741-3335/28/i=9A/a=014 http://stacks.iop.org/0741-3335/28/i=9A/a=014?key=crossref.2f7fb0fd445d2b2e047c5503ff107988}, -volume = {28}, -year = {1986} -} -@phdthesis{Karney1977b, -author = {Karney, Charles F. F.}, -file = {::}, -school = {MIT}, -title = {{Stochastic heating of Ions in a Tokamak by RF Power}}, -year = {1977} -} -@article{Jobes1985, -author = {Jobes, F. C. and Bernabei, S. and Chu, T. K. and Hooke, W. M. and Meservey, E. B. and Motley, R. W. and Stevens, J. E. and von Goeler, S.}, -doi = {10.1103/PhysRevLett.55.1295}, -file = {::}, -issn = {0031-9007}, -journal = {Physical Review Letters}, -month = {sep}, -number = {12}, -pages = {1295--1298}, -title = {{Current Rampup by Lower-Hybrid Waves in the PLT Tokamak}}, -url = {http://link.aps.org/doi/10.1103/PhysRevLett.55.1295}, -volume = {55}, -year = {1985} -} -@inproceedings{Bonoli2014, -abstract = {Progress in experiment and simulation capability in the lower hybrid range of frequencies at ITER relevant parameters is reviewed. Use of LH power in reactor devices is motivated in terms of its potential for efficient off-axis current profile control. Recent improvements in simulation capability including the development of full-wave field solvers, inclusion of the scrape off layer (SOL) in wave propagation codes, the use of coupled ray tracing/full-wave/3D (r v?, v//) Fokker Planck models, and the inclusion of wave scattering as well as nonlinear broadening effects in ray tracing / Fokker Planck codes are discussed. Experimental and modeling results are reviewed which are aimed at understanding the spectral gap problem in LH current drive (LHCD) and the density limit that has been observed and mitigated in LHCD experiments. Physics mechanisms that could be operative in these experiments are discussed, including toroidally induced variations in the parallel wavenumber, nonlinear broadening of the pump wave, scattering of LH waves from density fluctuations in the SOL, and spectral broadening at the plasma edge via full-wave effects.}, -author = {Bonoli, Paul T.}, -booktitle = {Physics of Plasmas}, -doi = {10.1063/1.4884360}, -file = {::;::}, -isbn = {9780735412101}, -issn = {15517616}, -keywords = {current drive,lower hybrid waves}, -pages = {15--24}, -title = {{Review of recent experimental and modeling progress in the lower hybrid range of frequencies at ITER relevant parameters}}, -url = {http://dx.doi.org/10.1063/1.4884360}, -volume = {21}, -year = {2014} -} -@article{Porkolab1984, -author = {Porkolab, M and Schuss, J J and Lloyd, B and Takase, Y and Texter, S and Bonoli, P. and Fiore, C and Gandy, R and Gwinn, D and Lipschultz, B and Marmar, E and Pappas, D and Parker, R and Pribyl, P}, -doi = {10.1103/PhysRevLett.53.450}, -issn = {0031-9007}, -journal = {Physical Review Letters}, -month = {jul}, -number = {5}, -pages = {450--453}, -publisher = {American Physical Society}, -title = {{Observation of Lower-Hybrid Current Drive at High Densities in the Alcator {\textless}math display="inline"{\textgreater} {\textless}mi{\textgreater}C{\textless}/mi{\textgreater} {\textless}/math{\textgreater} Tokamak}}, -url = {http://link.aps.org/doi/10.1103/PhysRevLett.53.450}, -volume = {53}, -year = {1984} -} -@article{Golant1972, -author = {Golant, V. E.}, -file = {::;::}, -journal = {Soviet Physics - Technical Physics}, -number = {12}, -pages = {1980--1988}, -title = {{Plasma Penetration Near The Lower Hybrid Frequency}}, -url = {papers2://publication/uuid/C77B2967-1EA5-4BC7-9523-8505AF8B295D}, -volume = {16}, -year = {1972} +@book{Jackson1998, +author = {Jackson, John David}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Ebooks/Electromagnetisme/Jackson/Jacksons Classical Electrodynamics (Solutions).pdf:pdf}, +publisher = {Wiley, New York}, +title = {{Classical Electrodynamics}}, +year = {1998} } @article{Litaudon1992a, abstract = {The TORE SUPRA lower hybrid current drive experiments (8 MW/3.7 GHz) use large phased waveguide arrays, four rows of 32 active waveguides and two passive waveguides for each of the two grills, to couple the waves to the plasma. These launchers are based on the 'multijunction' principle which allows them to be quite compact and is therefore attractive for the design of efficient multi-megawatt antennas in NET/ITER. Extensive coupling measurements have been performed in order to study the radiofrequency (RF) characteristics of the plasma loaded antennas. Measurements of the plasma scattering coefficients of the antennas show good agreement with those obtained from the linear coupling theory (SWAN code). Global reflection coefficients of a few per cent have been measured in a large range of edge plasma densities (0.3 x 10(18) m-3 less-than-or-equal-to n(eg) less-than-or-equal-to 1.4 x 10(18) m-3) or antenna positions (0.02-0.05 m from the plasma edge) and up to a maximum injected RF power density of 45 MW/m2. When the plasma is pushed against the inner wall of the chamber, the reflection coefficient is found to remain low up to distances of the order of 0. 10 m. The coupling measurements allow us to deduce the 'experimental' power spectra radiated by the antennas when all their modules are fed simultaneously with variable phases. Thus, the multijunction launcher is assessed as a viable antenna for high power transmission with good coupling characteristics and spectrum control.}, author = {Litaudon, X and Berger-by, G and Bibet, Philippe and Bizarro, Jo{\~{a}}o P.S. and Capitain, J J and Carrasco, J and Goniche, M and Hoang, G T and Kupfer, K and Magne, R. and Moreau, D and Peysson, Yves and Rax, J.-M. M and Rey, G and Rigaud, D and Tonon, G and Bergerby, G and Bibet, Philippe and Bizarro, Jo{\~{a}}o P.S. and Capitain, J J and Carrasco, J and Goniche, M and Hoang, G T and Kupfer, K and Magne, R. and Moreau, D and Peysson, Yves and Rax, J.-M. M and Rey, G and Rigaud, D and Tonon, G}, doi = {10.1088/0029-5515/32/11/I01}, -file = {::}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Litaudon1992.pdf:pdf}, isbn = {0029-5515}, issn = {00295515}, journal = {Nuclear Fusion}, @@ -125,83 +26,67 @@ @article{Litaudon1992a volume = {32}, year = {1992} } -@book{Mackay2010, -abstract = {The topics of anisotropy and bianisotropy are fundamental to electromagnetics from both theoretical and experimental perspectives. These properties underpin a host of complex and exotic electromagnetic phenomena in naturally occurring materials and in relativistic scenarios, as well as in artificially produced metamaterials. As a unique guide to this rapidly developing field, the book provides a unified presentation of key classic and recent results on the studies of constitutive relations, spacetime symmetries, planewave propagation, dyadic Green functions, and homogenization of composite materials. This book also offers an up-to-date extension to standard treatments of crystal optics with coverage on both linear and weakly nonlinear regimes.}, -author = {Mackay, Tom G and {Akhlesh Lakhtakia}}, -doi = {10.1111/ddi.12150}, -isbn = {9789814289610}, -pages = {1461--1467}, -title = {{Electromagnetic Anisotropy and Bianisotropy}}, -url = {http://www.worldscientific.com/doi/suppl/10.1142/7515/suppl{\_}file/7515{\_}chap01.pdf}, -year = {2010} -} -@book{Faria2008, -abstract = {The material presented in the book is built on a substrate of knowledge already provided by the basic sciences of mathematics and physics. Students are supposed to be acquainted with certain topics, such as linear algebra, differential equations, integral calculus, vector analysis and complex functions. If students still have difficulties with these topics, they may have to recap them in order to refresh their skills. This book is not a treatise on electricity and magnetism – its scope is far less ambitious. Its content can be delivered in a single-semester course, and is aimed to provide a scientifically founded and unified basis of fundamental knowledge on electromagnetic field phenomena that will help students follow up more advanced subjects covered in their courses. Topics are introduced in a systematic and friendly manner, proceeding from the simpler to more difficult ones, using a slow build-up process. In addition, a series of application examples and homework problems have been prepared to help students through the learning process. The fact that the book is partitioned into chapters does not imply that some of them can be skipped. Because the subject matter is deeply interrelated, students must try to adhere to the normal chapter sequence, otherwise they may be wasting their time or fail to get an integrated comprehensive view of the electromagnetic phenomena.}, -author = {Faria, J. A Brand{\~{a}}o}, -booktitle = {Electromagnetic Foundations of Electrical Engineering}, -doi = {10.1002/9780470697498}, -isbn = {9780470727096}, -pages = {1--399}, -title = {{Electromagnetic Foundations of Electrical Engineering}}, -year = {2008} +@article{Fisch1987, +author = {Fisch, Nathaniel J.}, +doi = {10.1103/RevModPhys.59.175}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Fisch1987{\_}Theory.of.Current.Drive.in.Plasmas.pdf:pdf}, +issn = {0034-6861}, +journal = {Rev. Mod. Phys.}, +month = {jan}, +number = {1}, +pages = {175--234}, +publisher = {American Physical Society}, +title = {{Theory of current drive in plasmas}}, +url = {http://link.aps.org/doi/10.1103/RevModPhys.59.175}, +volume = {59}, +year = {1987} } -@article{Karney1979c, +@article{Karney1978a, abstract = {The motion of an ion in a lower hybrid wave in a tokamak type plasma is studied. For ions with v⊥ ≥ $\omega$/k⊥, the motion is stochastic for fields satisfying E/B0 {\textgreater} ¼($\Omega$i /$\omega$)1/3($\omega$/k⊥). Provided that the perpendicular phase velocity, $\omega$/k⊥, can be slowed down to a few times the ion thermal speed, this stochastic ion motion may be an important mechanism by which injected rf power near the lower hybrid frequency can directly heat the ions.}, archivePrefix = {arXiv}, arxivId = {physics/0501034}, author = {Karney, Charles F. F.}, -doi = {10.1063/1.862512}, +doi = {10.1063/1.862406}, eprint = {0501034}, -file = {::}, +file = {:U$\backslash$:/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Karney1978{\_}Stochastic ion heating by a lower hybrid wave.pdf:pdf}, issn = {00319171}, journal = {Physics of Fluids}, keywords = {IONS,NONLINEAR PROBLEMS,PLASMA HEATING,PLASMA W}, -number = {11}, -pages = {2188}, +number = {9}, +pages = {1584}, primaryClass = {physics}, publisher = {AIP}, -title = {{Stochastic ion heating by a lower hybrid wave: II}}, -url = {http://scitation.aip.org/content/aip/journal/pof1/21/9/10.1063/1.862406 http://scitation.aip.org/content/aip/journal/pof1/22/11/10.1063/1.862512}, -volume = {22}, -year = {1979} -} -@article{Porkolab1984a, -author = {Porkolab, Miklos}, -doi = {10.1109/TPS.1984.4316303}, -file = {::}, -issn = {0093-3813}, -journal = {IEEE Transactions on Plasma Science}, -number = {2}, -pages = {107--117}, -title = {{Survey of Lower Hybrid Experiments}}, -url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=4316303}, -volume = {12}, -year = {1984} +title = {{Stochastic ion heating by a lower hybrid wave}}, +url = {https://www.zenodo.org/record/32013?ln=en http://scitation.aip.org/content/aip/journal/pof1/21/9/10.1063/1.862406}, +volume = {21}, +year = {1978} } -@book{Benford1992, -address = {Boston}, -author = {Benford, James and Swegle, John A}, -edition = {first edit}, -editor = {{Artech House}}, -isbn = {0890064156}, -pages = {412}, -title = {{High Power Microwaves}}, -year = {1992} +@inproceedings{Guilhem2009, +abstract = {The design and the fabrication of a new Lower Hybrid (LH) actively cooled antenna based on the passive active concept is a part of the CIMES project (Components for the Injection of Mater and Energy in Steady-state). The major objectives of Tore-Supra program is to achieve 1000 s pulses with this LH launcher, by coupling routinely {\textgreater}3 MW of LH wave at 3.7 GHz to the plasma with a parallel index n∥=1.7±0.2. The launcher is on its way to achieve its validation tests - low power Radio Frequency (RF) measurements, vacuum and hydraulic leak tests - and will be installed and commissioned on plasma during the fall of 2009. {\textcopyright} 2009 American Institute of Physics.}, +author = {Guilhem, D. and Others and Achard, Joelle and Belo, J. and Bertrand, Bernard and Bej, Z. and Bibet, Philippe and Brun, C. and Chantant, M. and Delmas, E. and Delpech, L{\'{e}}na and Doceul, Y. and Ekedahl, Annika and Goletto, C. and Goniche, M. and Hatchressian, J. C. and Hillairet, J. and Houry, M. and Joubert, P. and Lipa, M. and Madeleine, S. and Martinez, A. and Missirlian, M. and Poli, S. and Portafaix, C. and Raulin, D. and Saille, A. and Soler, B. and Thouvenin, D. and Verger, Jean-Marc and Vulliez, Karl and Zago, B. and Bobkov, Volodymyr and Noterdaeme, Jean-Marie}, +booktitle = {Radio Frequency Power In Plasmas: Proceedings Of The 18th Topical Conference, Aip Conference Proceedings}, +doi = {10.1063/1.3273786}, +isbn = {9780735407534}, +issn = {0094243X 15517616}, +keywords = {ITER,Lower hybrid,Passive-active,Tore,[Brazing}, +pages = {435--438}, +title = {{Passive Active Multi-Junction 3.7 {\{}GHz{\}} Launcher for Tore-Supra Long Pulse Experiments : Manufacturing Process and Tests}}, +url = {http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.3273786}, +volume = {1187}, +year = {2009} } -@article{Ridolfini2005, -abstract = {A prototype passive active multijunction (PAM) launcher for lower hybrid (LH) waves conceptually similar to that foreseen for ITER has been successfully tested on FTU at frequency f = 8 GHz. The power routinely and safely managed for the maximum time allowed by the LH power plant (0.9 s) without any fault in the transmission lines is 250 kW, corresponding to 75 MW m ?2 across the antenna active area and very close to the design value of 270 kW or 80 MW m ?2 . The achieved value is at least 1.4 times larger than the ITER request, which would be only 52 MW m ?2 , if the 33 MW m ?2 required to the ITER grill in order to couple 20 MW to the plasma, are scaled up linearly with f , from f ITER = 5 GHz. This linear scaling of the power handling capability of the LH antennae is indeed conservative with respect to the available data. The test results validate also the other two main expectations relevant to ITER, foreseen by the codes, namely to operate with the grill entirely in the vessel shadow and to still preserve good current drive (CD) efficiency. Even with the grill mouth retracted 2 mm inside the port shadow and with density in front of the launcher very close or even lower than the cut-off value, the PAM reflection coefficient is always ? 2.5{\%}, if the antenna has been properly conditioned. The CD efficiency is comparable to that of a conventional grill, once the lower directivity is taken into account. Flexibility in determining the N || spectrum is also maintained, according to hard x-rays and electron cyclotron emission spectra. Conditioning the PAM in order to operate at the ITER equivalent power level required only one day of radio-frequency operation, without a previous baking of the waveguides.}, -author = {Ridolfini, V Pericoli and Bibet, Philippe and Mirizzi, F and Apicella, M.L L and Barbato, E and Buratti, P. and Calabr{\`{o}}, G and Cardinali, A and Granucci, G and Panaccione, L and Podda, S and Sozzi, C and Tuccillo, A.A A and Calabr�, G and Cardinali, A and Granucci, G and Panaccione, L and Podda, S and Sozzi, C and Tuccillo, A.A A}, -doi = {10.1088/0029-5515/45/9/008}, -file = {::;::}, -issn = {0029-5515}, -journal = {Nuclear Fusion}, -month = {sep}, -number = {9}, -pages = {1085}, -title = {{LHCD and coupling experiments with an ITER-like PAM launcher on the FTU tokamak}}, -url = {http://stacks.iop.org/0029-5515/45/i=9/a=008 http://stacks.iop.org/0029-5515/45/i=9/a=008?key=crossref.6b9bde59e11fd1ed8c9b8be441b530ec}, -volume = {45}, -year = {2005} +@article{Krapchev1979, +author = {Vladimir, Krapchev and Krapchev, Vladimir B.}, +doi = {10.1103/PhysRevLett.42.497}, +issn = {0031-9007}, +journal = {Physical Review Letters}, +month = {feb}, +number = {8}, +pages = {497--500}, +title = {{Kinetic Thoery of the Pondermotive Effects in Plasma}}, +url = {https://link.aps.org/doi/10.1103/PhysRevLett.42.497}, +volume = {42}, +year = {1979} } @article{Hooke1972, author = {Hooke, W M and Bernabei, S.}, @@ -215,44 +100,26 @@ @article{Hooke1972 volume = {28}, year = {1972} } -@article{Ekedahl2010b, -abstract = {A new ITER-relevant lower hybrid current drive (LHCD) launcher, based on the passive-active-multijunction (PAM) concept, was brought into operation on the Tore Supra tokamak in autumn 2009. The PAM launcher concept was designed in view of ITER to allow efficient cooling of the waveguides, as required for long pulse operation. In addition, it offers low power reflection close to the cut-off density, which is very attractive for ITER, where the large distance between the plasma and the wall may bring the density in front of the launcher to low values. The first experimental campaign on Tore Supra has shown extremely encouraging results in terms of reflected power level and power handling. Power reflection coefficient {\textless}2{\%} is obtained at low density in front of the launcher, i.e. close to the cut-off density, and very good agreement between the experimental results and the coupling code predictions is obtained. Long pulse operation at ITER-relevant power density has been demonstrated. The maximum power and energy reached so far is 2.7MW during 78 s, corresponding to a power density of 25MWm-2, i.e. its design value at f = 3.7 GHz. In addition, 2.7MWhas been coupled at a plasma-launcher distance of 10 cm, with a power reflection coefficient {\textless}2{\%}. Finally, full non-inductive discharges have been sustained for 50 s with the PAM. {\textcopyright} 2010 IOP Publishing Ltd.}, -author = {Ekedahl, Annika and Delpech, L{\'{e}}na and Goniche, M. and Guilhem, D. and Hillairet, J. and Preynas, M{\'{e}}lanie and Sharma, P.K. and Achard, Joelle and Bae, Y.S. S. and Bai, X. and Balorin, C. and Baranov, Y. and Basiuk, V. and B{\'{e}}coulet, A. and Belo, J. and Berger-By, G. and Bremond, S. and Castaldo, C. and Ceccuzzi, Silvio and Cesario, Roberto and Corbel, E. and Courtois, X. and Decker, Joan and Delmas, E. and Ding, X. and Douai, D. and Goletto, C. and Gunn, J.P. and Hertout, P. and Hoang, G.T. and Imbeaux, F. and Kirov, K.K. and Litaudon, X. and Magne, R. and Mailloux, Joelle and Mazon, D. and Mirizzi, F. and Mollard, Patrick and Moreau, P. and Oosako, T. and Petr{\v{z}}{\'{i}}lka, V. A. and Peysson, Yves and Poli, S. and Prou, Marc and Saint-Laurent, F. and Samaille, F. and Saoutic, B. and Others}, -doi = {10.1088/0029-5515}, -issn = {00295515 17414326}, -journal = {Nuclear Fusion}, -number = {11}, -title = {{Validation of the ITER-relevant passive-active-multijunction LHCD launcher on long pulses in Tore Supra}}, -volume = {50}, -year = {2010} -} -@article{ITERPhysics_Chap6, -author = {Drive, ITER Physics Expert Group on Energe and Editors, ITER Physics Basis}, -doi = {10.1088/0029-5515/40/7/512}, -file = {::}, -issn = {0029-5515}, -journal = {Nuclear Fusion}, -month = {jul}, -number = {7}, -pages = {1429--1429}, -title = {{ITER Physics Basis Chapter 6: Plasma auxiliary heating and current drive}}, -url = {http://stacks.iop.org/0029-5515/40/i=7/a=512?key=crossref.76a491e853c938c421694a29456975fe}, -volume = {40}, -year = {2000} +@article{Gormezano1986a, +author = {Gormezano, C}, +doi = {10.1088/0741-3335/28/9A/014}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Gormezano1986{\_}Review{\_}LHCD.pdf:pdf}, +issn = {0741-3335}, +journal = {Plasma Physics and Controlled Fusion}, +month = {sep}, +number = {9A}, +pages = {1365--1376}, +title = {{Review of lower hybrid wave heating and current drive}}, +url = {http://stacks.iop.org/0741-3335/28/i=9A/a=014?key=crossref.2f7fb0fd445d2b2e047c5503ff107988}, +volume = {28}, +year = {1986} } -@book{Swanson2003, -author = {Swanson, D Gary}, -isbn = {0 7503 0927 X}, -publisher = {Taylor {\&} Francis, 2nd edition, Bristol}, -title = {{Plasma Waves}}, -year = {2003} -} -@techreport{Hooke1982, -author = {Hooke, W and Others}, -file = {::}, -institution = {PPPL-1976}, -title = {{Lower Hybrid Heating and Current Drive on {\{}PLT{\}}}}, -year = {1983} +@inproceedings{Kim2012, +author = {Kim, K and Kim, H K T and Kim, H K T and Park, S and Bae, Y.S. S. and Yang, H L}, +booktitle = {KSTAR conference in Muju}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Kim2012{\_}fabrication of KSTAR 5 GHz LHCD launcher coupler.pdf:pdf}, +title = {{Fabrication of KSTAR 5 GHz LHCD launcher coupler}}, +year = {2012} } @article{Bellan1974, author = {Bellan, P M and Porkolab, M}, @@ -264,72 +131,53 @@ @article{Bellan1974 volume = {17}, year = {1974} } -@article{Krapchev1981, -author = {Krapchev, V B and Theilhaber, K S and Ko, K C and Bers, A}, -doi = {10.1103/PhysRevLett.46.1398}, -issn = {0031-9007}, -journal = {Physical Review Letters}, -month = {may}, -number = {21}, -pages = {1398--1401}, -publisher = {American Physical Society}, -title = {{Nonlinear Coupling of Lower-Hybrid Waves at the Edge of Tokamak Plasmas}}, -url = {http://link.aps.org/doi/10.1103/PhysRevLett.46.1398}, -volume = {46}, -year = {1981} -} -@article{Gormezano1984a, -abstract = {The experimental data of the HF coupling measurements of the 4 waveguide grill of the Wega Tokamak have been compared with a linear coupling theory using a step density model. In order to minimize specific boundary effects which are not taken into account in the theory, the authors made use of data obtained when only the central waveguides are fed. The plasma density and its gradient at the mouth of the grill are estimated from probe measurements made with plasma conditions similar to those of the experimental coupling data. The qualitative agreement is always very good and a quantitative agreement is obtained in a relatively high density regime. The validity of the step density model is supported by the density dependence of the phases of the reflected signals.}, -author = {Gormezano, C. and Moreau, D}, -doi = {10.1088/0741-3335/26/3/005}, -file = {::}, -isbn = {0741-3335}, -issn = {0741-3335}, -journal = {Plasma Physics and Controlled Fusion}, -month = {mar}, -number = {3}, -pages = {553}, -title = {{Lower hybrid wave coupling in the Wega Tokamak}}, -url = {http://stacks.iop.org/0741-3335/26/i=3/a=005 http://stacks.iop.org/0741-3335/26/i=3/a=005?key=crossref.b0988de8436642feb6abfe45dbf08c33}, -volume = {26}, -year = {1984} -} -@article{Krapchev1979, -author = {Vladimir, Krapchev and Krapchev, Vladimir B.}, -doi = {10.1103/PhysRevLett.42.497}, -issn = {0031-9007}, -journal = {Physical Review Letters}, -month = {feb}, -number = {8}, -pages = {497--500}, -title = {{Kinetic Thoery of the Pondermotive Effects in Plasma}}, -url = {https://link.aps.org/doi/10.1103/PhysRevLett.42.497}, -volume = {42}, -year = {1979} +@book{Stix1992, +author = {Stix, Thomas Howard}, +isbn = {978-0-88318-859-0}, +publisher = {Springer, New-York}, +title = {{Waves in Plasmas}}, +year = {1992} } -@article{Motley1980, -author = {Motley, R W and Hooke, W M}, -doi = {10.1088/0029-5515/20/2/013}, -file = {::;::}, -issn = {17414326}, -journal = {Nuclear Fusion}, -pages = {222--224}, -title = {{Active-passive waveguide array for wave excitation in plasmas}}, -volume = {222}, +@article{Karney1979c, +abstract = {The motion of an ion in a lower hybrid wave in a tokamak type plasma is studied. For ions with v⊥ ≥ $\omega$/k⊥, the motion is stochastic for fields satisfying E/B0 {\textgreater} ¼($\Omega$i /$\omega$)1/3($\omega$/k⊥). Provided that the perpendicular phase velocity, $\omega$/k⊥, can be slowed down to a few times the ion thermal speed, this stochastic ion motion may be an important mechanism by which injected rf power near the lower hybrid frequency can directly heat the ions.}, +archivePrefix = {arXiv}, +arxivId = {physics/0501034}, +author = {Karney, Charles F. F.}, +doi = {10.1063/1.862512}, +eprint = {0501034}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Karney1978{\_}Stochastic ion heating by a lower hybrid wave.pdf:pdf}, +issn = {00319171}, +journal = {Physics of Fluids}, +keywords = {IONS,NONLINEAR PROBLEMS,PLASMA HEATING,PLASMA W}, +number = {11}, +pages = {2188}, +primaryClass = {physics}, +publisher = {AIP}, +title = {{Stochastic ion heating by a lower hybrid wave: II}}, +url = {http://scitation.aip.org/content/aip/journal/pof1/21/9/10.1063/1.862406 http://scitation.aip.org/content/aip/journal/pof1/22/11/10.1063/1.862512}, +volume = {22}, year = {1979} } -@inproceedings{Meneghini2010b, -author = {Meneghini, Orso and Shiraiwa, Syunichi and Johnson, David and Faust, I.C. and Kanojia, Atma and Parker, Ronald and Terry, David and Vieira, Rui and Wallace, Gregory M and {Wilson Randy; Wukitch}, Steve}, -booktitle = {American Physical Society, 52nd Annual Meeting of the APS Division of Plasma Physics}, -month = {nov}, -title = {{Coupling of LH waves with four-way-splitter antenna on Alcator C-Mod}}, -year = {2010} +@inproceedings{Bonoli2014, +abstract = {Progress in experiment and simulation capability in the lower hybrid range of frequencies at ITER relevant parameters is reviewed. Use of LH power in reactor devices is motivated in terms of its potential for efficient off-axis current profile control. Recent improvements in simulation capability including the development of full-wave field solvers, inclusion of the scrape off layer (SOL) in wave propagation codes, the use of coupled ray tracing/full-wave/3D (r v?, v//) Fokker Planck models, and the inclusion of wave scattering as well as nonlinear broadening effects in ray tracing / Fokker Planck codes are discussed. Experimental and modeling results are reviewed which are aimed at understanding the spectral gap problem in LH current drive (LHCD) and the density limit that has been observed and mitigated in LHCD experiments. Physics mechanisms that could be operative in these experiments are discussed, including toroidally induced variations in the parallel wavenumber, nonlinear broadening of the pump wave, scattering of LH waves from density fluctuations in the SOL, and spectral broadening at the plasma edge via full-wave effects.}, +author = {Bonoli, Paul T.}, +booktitle = {Physics of Plasmas}, +doi = {10.1063/1.4884360}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Bonoli2014.pdf:pdf;:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Bonoli2014{\_}Review recent experimental and modeling progress in the lower hybrid range of frequencies at ITER relevant parameters{\_}PhysPlasma.pdf:pdf}, +isbn = {9780735412101}, +issn = {15517616}, +keywords = {current drive,lower hybrid waves}, +pages = {15--24}, +title = {{Review of recent experimental and modeling progress in the lower hybrid range of frequencies at ITER relevant parameters}}, +url = {http://dx.doi.org/10.1063/1.4884360}, +volume = {21}, +year = {2014} } @article{Theilhaber1980, abstract = {Launching of the fast wave in the lower hybrid frequency range is described. This wave is excited at the plasma edge by RF electric fields perpendicular to those required for the lower hybrid wave. In high-temperature plasmas, where the lower hybrid wave may not penetrate because of Landau damping or other effects near the edge, the fast wave might provide an alternative for heating and/or current generation in the central portion of the plasma. In addition, for high-density plasmas, this has the advantage that lower frequencies than those required for the lower hybrid excitation can be used. Thus waveguides of convenient dimensions for maximum power transmission and ease of fabrication can be employed. Coupling from a waveguide array into an inhomogeneous plasma is analysed. The model is infinite in y, the direction perpendicular to magnetic field and density gradient. Power reflection in the waveguides is found as a function of array design and density gradient at the edge. This reflection is fairly large ({\textgreater} 20{\%}). Propagation into the plasma is then considered, and the field structure and dispersion of the fast waves are found as functions of the distance of penetration. Unlike the lower hybrid waves, fast waves do not form resonance cones and energy is dispersed over a large volume.}, author = {Theilhaber, K and Bers, A}, doi = {10.1088/0029-5515/20/5/003}, -file = {::}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Theilhaber1980{\_}Coupling to the Fast Wave at LH frequencies.pdf:pdf}, issn = {0029-5515}, journal = {Nuclear Fusion}, month = {may}, @@ -340,32 +188,105 @@ @article{Theilhaber1980 volume = {20}, year = {1980} } -@article{Tonon1984, -author = {Tonon, G}, -doi = {10.1088/0741-3335/26/1A/313}, -file = {::}, -issn = {0741-3335}, -journal = {Plasma Physics and Controlled Fusion}, -month = {jan}, -number = {1A}, -pages = {145--155}, -title = {{Heating and current drive by LH-Waves on toroidal plasmas: problems and perspectives}}, -url = {http://stacks.iop.org/0741-3335/26/i=1A/a=313?key=crossref.48a2b98169970f4f579f24dc8c805348 http://stacks.iop.org/0741-3335/26/i=1A/a=313}, -volume = {26}, -year = {1984} +@book{Kolundzija2002, +author = {Kolund{\v{z}}ija, Branko and Djordjevi{\'{c}}, A.}, +isbn = {0890063605}, +pages = {408}, +publisher = {Artech House}, +title = {{Electromagnetic Modeling of Composite Metallic and Dielectric Structures}}, +year = {2002} } -@misc{Tenenbaum2003, -author = {Tenenbaum, Peter}, -file = {::}, -title = {{A Brief Introduction to RF Power Sources Klystrons}}, -url = {http://www.desy.de/{~}njwalker/uspas/}, -year = {2003} +@article{Brambilla1979, +abstract = {... The theory of waveguide launching of lower hybrid (LH) waves has been developed in Refs [1-5] and found to be in satisfactory agreement with ... the fraction of power able to penetrate into the plasma core essentially depends on the value of the critical index for accessibility , Eq.(l ...}, +author = {Brambilla, Marco}, +doi = {10.1088/0029-5515/19/10/006}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Brambilla1979.pdf:pdf}, +issn = {0029-5515}, +journal = {Nuclear Fusion}, +month = {oct}, +number = {10}, +pages = {1343--1357}, +title = {{Waveguide launching of lower hybrid waves}}, +url = {http://stacks.iop.org/0029-5515/19/i=10/a=006?key=crossref.f75bd0add10137b71a4d878a71971b22}, +volume = {19}, +year = {1979} +} +@article{Tonon1982, +author = {Tonon, G and Others}, +journal = {International Atomic Energy Agency (IAEA): IAEA.}, +title = {{Lower Hybrid Current Drive and heating on the {\{}WEGA{\}} Tokamak}}, +year = {1983} +} +@inproceedings{Motley1985, +abstract = {A study of radiofrequency current ramp-up in the PLT tokamak is reported. The plasma current was first raised to 200–300 kA by the Ohmic heating transformer, and the current in the transformer primary circuit was then held constant to remove the OH drive. After the current fell below 200 kA, up to 300 kW of toroidally-directed RF power at 800 MHz was transmitted into the PLT plasma via a 6-element phased waveguide array. Current ramp-up rates between 0 and 120 kA/s for a 0.35 s time interval ((½–1/3) L/R time) where measured at densities between 2 and 4 × 1012 cm−3. It is estimated that about 20{\%} of the RF energy introduced into the vacuum vessel was converted into poloidal magnetic field energy, LI2/2, where L ≅ 3 $\mu$H is the total inductance of the plasma current loop. This conversion ratio should depend on a variety of factors, including the percentage of RF power absorbed by resonant electrons and the magnitude of the back current induced by the changing poloidal flux LI. The high ramp-up efficiencies are predicted theoretically in the regime in which the PLT ramp-up experiments operate, i.e., where the phase velocity of the waves is approximately equal in magnitude to the runaway velocity due to the back voltage. Comparison of the raw data with theory suggest that about ½ to ¾ of the incident RF power is absorbed by resonant high-velocity electrons.}, +annote = {Paper IAEA--CN--44/F--II--2 +Proc. Tenth International Conf., London, England, Sept. 12--19, 1984}, +author = {Motley, Robert W and Bell, Ronald E and Bernabei, Stefano and Cavallo, Alfred J and Chu, Tsu-Kai and Cohen, Samuel A and Denne, Boel G and Efthimion, Phillip C and Fisch, Nathaniel J. and Hinnov, Einar and Hooke, William M and Hosea, Joel C and Jobes, Forrest C and Karney, Charles F. F. and Mazzucato, Ernesto and Meservey, E. and Stevens, James E and Suckewer, Szymon and Taylor, Gary and Timberlake, John R and von Goeler, Schweickhard E and Wilson, J Randall}, +booktitle = {Plasma Physics and Controlled Nuclear Fusion Research 1984}, +pages = {473--478}, +publisher = {IAEA, Vienna}, +title = {{Lower Hybrid Current Ramp-up in the {\{}PLT{\}} Tokamak}}, +url = {https://inis.iaea.org/search/search.aspx?orig{\_}q=RN:16032336}, +volume = {1}, +year = {1985} +} +@book{Benford2015, +author = {Benford, James and Swegle, John A and Schamiluglu, Edl}, +edition = {third edit}, +isbn = {9781482260595}, +keywords = {Multipactor}, +mendeley-tags = {Multipactor}, +pages = {453}, +publisher = {CRC Press}, +title = {{High Power Microwaves}}, +year = {2015} +} +@book{Lindell1995, +author = {Lindell, Ismo V}, +editor = {Press, IEEE}, +isbn = {978-0198562399}, +title = {{Methods for Electromagnetic Field Analysis}}, +year = {1995} +} +@book{Pozar1998, +author = {Pozar, David M.}, +edition = {3rd}, +editor = {Wiley}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Ebooks/Electromagnetisme/Pozar/3rd{\_}Edition/550{\_}2{\_}contents.pdf:pdf}, +isbn = {0471448788}, +publisher = {John Wiley {\&} Sons, Inc.}, +title = {{Microwave Engineering}}, +year = {1998} +} +@article{TONON1977, +author = {TONON, G. and BLANC, P. and GORMEZANO, C. and HESS, W. and ICHTCHENKO, G. and NGUYEN, T. K. and DURVAUX, M. and MAGNE, R. and OHLENDORF, W. and PACHER, G. and PACHER, H. and WEGROWE, J. G.}, +doi = {10.1051/jphyscol:1977615}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Tonon1977.pdf:pdf}, +issn = {0449-1947}, +journal = {Le Journal de Physique Colloques}, +month = {dec}, +number = {C6}, +pages = {C6--161--C6--169}, +title = {{LOWER HYBRID FREQUENCY HEATING IN TOROIDAL DEVICES WITH EMPHASIS ON WEGA RESULTS}}, +url = {http://www.edpsciences.org/10.1051/jphyscol:1977615 http://dx.doi.org/10.1051/jphyscol:1977615}, +volume = {38}, +year = {1977} +} +@article{Jackson1999, +author = {Jackson, J.D. and Wijk, Kasper}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Ebooks/Electromagnetisme/Jackson/Jacksons Classical Electrodynamics (Solutions).pdf:pdf}, +journal = {American Journal of Physics}, +pages = {841}, +title = {{Classical Electrodynamics-Solutions}}, +url = {http://link.aip.org/link/?AJPIAS/67/841/27 )Z7}, +volume = {67}, +year = {1999} } @article{Bibet1995, abstract = {A new concept-the reflector waveguide array-is proposed to improve and simplify the design of continuous wave lower hybrid (LH) launchers for steady state reactor applications. Mechanical robustness of the antenna and efficient heat removal are provided by a thick wall waveguide structure that can accommodate a large number of cooling ducts. The plasma facing mouthpiece could be made of the same material as the reactor first wall and could be easily replaceable through remote handling. In order to compensate for the increased horizontal distance between adjacent waveguides, the front ends of the thick septa are grooved to form short ( equivalent to lambda /4) passive waveguides that act as reflectors between the radiofrequency powered waveguides (drivers). Then, for an adequate phasing between the active waveguides, the total electric field at the reflector waveguide apertures varies coherently with the one in the drivers to launch a highly directional slow wave. It is shown that the coupling properties of such an array and the directivity of the radiated power spectrum are similar to those of present day launchers. Their dependences upon the depth of the reflector waveguides, and the electron density and its gradient are investigated. The effect of changing the phase between the drivers is also studied. The proposed reflector LH antenna would provide enough flexibility to vary the N // spectrum for plasma control purposes in a steady state fusion reactor}, author = {Bibet, Philippe and Litaudon, Xavier and Moreau, Didier}, doi = {10.1088/0029-5515/35/10/I05}, -file = {::}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Bibet1995.pdf:pdf}, isbn = {0029-5515}, issn = {0029-5515}, journal = {Nuclear Fusion}, @@ -376,88 +297,142 @@ @article{Bibet1995 volume = {35}, year = {1995} } -@article{Search, -author = {Drive, ITER Physics Expert Group on Energe and Editors, ITER Physics Basis}, -doi = {10.1088/0029-5515/40/7/512}, -file = {::}, -issn = {0029-5515}, -journal = {Nuclear Fusion}, -month = {jul}, -number = {7}, -pages = {1429--1429}, -title = {{ITER Physics Basis Chapter 6: Plasma auxiliary heating and current drive}}, -url = {http://stacks.iop.org/0029-5515/40/i=7/a=512?key=crossref.76a491e853c938c421694a29456975fe}, -volume = {40}, -year = {2000} +@techreport{Moreau1984, +author = {Moreau, Didier and Nguyen, T K}, +institution = {{\{}EUR-CEA-FC1246{\}} Euratom-{\{}CEA{\}}}, +title = {{Couplage de l'onde lente au voisinage de la fr{\'{e}}quence hybride basse dans les grands Tokamaks}}, +year = {1984} } -@inproceedings{Kim2012, -author = {Kim, K and Kim, H K T and Kim, H K T and Park, S and Bae, Y.S. S. and Yang, H L}, -booktitle = {KSTAR conference in Muju}, -file = {::}, -title = {{Fabrication of KSTAR 5 GHz LHCD launcher coupler}}, -year = {2012} +@article{Warnick2014, +author = {Warnick, Karl F. and Russer, Peter H.}, +doi = {10.2528/PIER14063009}, +issn = {1559-8985}, +journal = {Progress In Electromagnetics Research}, +pages = {83--112}, +title = {{DIFFERENTIAL FORMS AND ELECTROMAGNETIC FIELD THEORY}}, +url = {http://www.jpier.org/PIER/pier.php?paper=14063009}, +volume = {148}, +year = {2014} } -@phdthesis{Meneghini2012, -author = {Meneghini, Orso}, -school = {Massachusetts Institute of Technology}, -title = {{Full-Wave modeling of lower hybrid waves on Alcator C-Mod}}, -year = {2012} +@book{Griffiths2005, +abstract = {Features a clear, accessible treatment of the fundamentals of electromagnetic theory. Its lean and focused approach employs numerous examples and problems. Carefully discusses subtle or difficult points. Contains numerous, relevant problems within the book in addition to end of each chapter problems and answers.}, +archivePrefix = {arXiv}, +arxivId = {0712.0689}, +author = {Griffiths, David J. and Inglefield, Colin}, +booktitle = {American Journal of Physics}, +doi = {10.1119/1.4766311}, +eprint = {0712.0689}, +isbn = {978-0138053260}, +issn = {00029505}, +pages = {574}, +pmid = {20560239}, +title = {{Introduction to Electrodynamics}}, +volume = {73}, +year = {2005} } -@article{Fisch1987, -author = {Fisch, Nathaniel J.}, -doi = {10.1103/RevModPhys.59.175}, -file = {::}, -issn = {0034-6861}, -journal = {Rev. Mod. Phys.}, -month = {jan}, -number = {1}, -pages = {175--234}, +@article{Briggs1972, +author = {Briggs, R. J. and Parker, Ronald R}, +doi = {10.1103/PhysRevLett.29.852}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Briggs1972{\_}Transport of rf Energy to the Lower Hybrid Resonance in an Inhomogeneous Plasma.pdf:pdf}, +issn = {0031-9007}, +journal = {Phys. Rev. Lett.}, +month = {sep}, +number = {13}, +pages = {852--855}, publisher = {American Physical Society}, -title = {{Theory of current drive in plasmas}}, -url = {http://link.aps.org/doi/10.1103/RevModPhys.59.175}, -volume = {59}, -year = {1987} -} -@book{Stix1992, -author = {Stix, Thomas Howard}, -isbn = {978-0-88318-859-0}, -publisher = {Springer, New-York}, -title = {{Waves in Plasmas}}, -year = {1992} +title = {{Transport of rf Energy to the Lower Hybrid Resonance in an Inhomogeneous Plasma}}, +url = {http://link.aps.org/doi/10.1103/PhysRevLett.29.852}, +volume = {29}, +year = {1972} } -@article{Litaudon1990a, -author = {Litaudon, X and Moreau, D}, -doi = {10.1088/0029-5515/30/3/009}, +@article{Schuss1981, +author = {Schuss, J.J. and Porkolab, M. and Takase, Y. and Cope, D. and Fairfax, S. and Greenwald, M. and Gwinn, D. and Hutchinson, I.H. and Kusse, B. and Marmar, E. and Overskei, D. and Pappas, D. and Parker, R.R. and Scaturro, L. and West, J. and Wolfe, S.}, +doi = {10.1088/0029-5515/21/4/001}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Schuss1981{\_}Lower Hybrid Heating in the Alcator A tokamak.pdf:pdf}, issn = {0029-5515}, journal = {Nuclear Fusion}, -month = {mar}, -number = {3}, -pages = {471--484}, -title = {{Coupling of slow waves near the lower hybrid frequency in JET}}, -url = {http://stacks.iop.org/0029-5515/30/i=3/a=009?key=crossref.1b8efc2102e93db4c4a8a0cdc2275c12}, -volume = {30}, -year = {1990} +month = {apr}, +number = {4}, +pages = {427--452}, +title = {{Lower hybrid heating in the Alcator A tokamak}}, +url = {https://www.google.com/url?sa=t{\&}rct=j{\&}q={\&}esrc=s{\&}source=web{\&}cd=1{\&}cad=rja{\&}uact=8{\&}ved=0ahUKEwjg-66-19zOAhXCORQKHbmzC1kQFggcMAA{\&}url=http{\%}3A{\%}2F{\%}2Fdspace.mit.edu{\%}2Fbitstream{\%}2Fhandle{\%}2F1721.1{\%}2F94663{\%}2F80ja011{\_}full.pdf{\%}3Fsequence{\%}3D1{\&}usg=AFQjCNGhibBkLEgGPQ3RWNH}, +volume = {21}, +year = {1981} } -@article{Theilhaber1980, -abstract = {Launching of the fast wave in the lower hybrid frequency range is described. This wave is excited at the plasma edge by RF electric fields perpendicular to those required for the lower hybrid wave. In high-temperature plasmas, where the lower hybrid wave may not penetrate because of Landau damping or other effects near the edge, the fast wave might provide an alternative for heating and/or current generation in the central portion of the plasma. In addition, for high-density plasmas, this has the advantage that lower frequencies than those required for the lower hybrid excitation can be used. Thus waveguides of convenient dimensions for maximum power transmission and ease of fabrication can be employed. Coupling from a waveguide array into an inhomogeneous plasma is analysed. The model is infinite in y, the direction perpendicular to magnetic field and density gradient. Power reflection in the waveguides is found as a function of array design and density gradient at the edge. This reflection is fairly large ({\textgreater} 20{\%}). Propagation into the plasma is then considered, and the field structure and dispersion of the fast waves are found as functions of the distance of penetration. Unlike the lower hybrid waves, fast waves do not form resonance cones and energy is dispersed over a large volume.}, -author = {Theilhaber, K and Bers, A}, -doi = {10.1088/0029-5515/20/5/003}, -file = {::}, +@article{Peysson2012, +abstract = {A new ray-tracing code named C3P O has been developed to study the propagation of arbitrary electromagnetic radio-frequency (rf) waves in magnetized toroidal plasmas. Its structure is designed for maximum flexibility regarding the choice of coordinate system and dielectric model. The versatility of this code makes it particularly suitable for integrated modeling systems. Using a coordinate system that reflects the nested structure of magnetic flux surfaces in tokamaks, fast and accurate calculations inside the plasma separatrix can be performed using analytical derivatives of a spline-Fourier interpolation of the axisymmetric toroidal MHD equilibrium. Applications to reverse field pinch magnetic configuration are also included. The effects of 3D perturbations of the axisymmetric toroidal MHD equilibrium, due to the discreteness of the magnetic coil system or plasma fluctuations in an original quasi-optical approach, are also studied. Using a Runge–Kutta–Fehlberg method for solving the set of ordinary differential equations, the ray-tracing code is extensively benchmarked against analytical models and other codes for lower hybrid and electron cyclotron waves. (Some figures may appear in colour only in the online journal)}, +author = {Peysson, Yves and Decker, Joan and Morini, L}, +doi = {10.1088/0741-3335/54/4/045003}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Peysson2012{\_}A versatile ray-tracing code for studying rf wave propagation in toroidal magnetized plasmas.pdf:pdf;:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Peysson2012{\_}A versatile rayTracing code for studying rf wave propagation in toroidal magnetized plasma.pdf:pdf}, +issn = {0741-3335}, +journal = {Plasma Phys. Control. Fusion}, +number = {54}, +pages = {45003--45003}, +title = {{A versatile ray-tracing code for studying rf wave propagation in toroidal magnetized plasmas}}, +url = {http://iopscience.iop.org/0741-3335/54/4/045003}, +volume = {54}, +year = {2012} +} +@book{Harrington2001, +author = {Harrington, Roger F}, +isbn = {978-0-471-20806-8}, +pages = {496}, +publisher = {Wiley-IEEE Press, New York}, +title = {{Time-Harmonic Electromagnetic Fields}}, +year = {2001} +} +@book{Smith1997, +abstract = {Basic theory of classical electromagnetism -- Electromagnetic plane waves in free space: polarized waves -- Inhomogeneous plane waves and the plane-wave spectrum -- Electromagnetic analoques of some optical principles -- Radiation from distributions of charge and current: general formulation -- Electromagnetic field of a moving point charge -- Dipole radiation -- Radiation from thin-wire antennas.}, +author = {Smith, Glenn S.}, +isbn = {0521586984}, +pages = {653}, +title = {{An introduction to classical electromagnetic radiation}}, +url = {https://books.google.com/books?id=m8RzbqS772IC{\&}pg=PA474}, +year = {1997} +} +@article{Porkolab1984, +author = {Porkolab, M and Schuss, J J and Lloyd, B and Takase, Y and Texter, S and Bonoli, P. and Fiore, C and Gandy, R and Gwinn, D and Lipschultz, B and Marmar, E and Pappas, D and Parker, R and Pribyl, P}, +doi = {10.1103/PhysRevLett.53.450}, +issn = {0031-9007}, +journal = {Physical Review Letters}, +month = {jul}, +number = {5}, +pages = {450--453}, +publisher = {American Physical Society}, +title = {{Observation of Lower-Hybrid Current Drive at High Densities in the Alcator {\textless}math display="inline"{\textgreater} {\textless}mi{\textgreater}C{\textless}/mi{\textgreater} {\textless}/math{\textgreater} Tokamak}}, +url = {http://link.aps.org/doi/10.1103/PhysRevLett.53.450}, +volume = {53}, +year = {1984} +} +@article{Search, +author = {Drive, ITER Physics Expert Group on Energe and Editors, ITER Physics Basis}, +doi = {10.1088/0029-5515/40/7/512}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/ITER physics basis{\_}Heating and Current Drive.pdf:pdf}, issn = {0029-5515}, journal = {Nuclear Fusion}, -month = {may}, -number = {5}, -pages = {547--555}, -title = {{Coupling to the fast wave at lower hybrid frequencies}}, -url = {http://stacks.iop.org/0029-5515/20/i=5/a=003?key=crossref.d24dafab0e5f71c54c1f342307d0b320}, -volume = {20}, -year = {1980} +month = {jul}, +number = {7}, +pages = {1429--1429}, +title = {{ITER Physics Basis Chapter 6: Plasma auxiliary heating and current drive}}, +url = {http://stacks.iop.org/0029-5515/40/i=7/a=512?key=crossref.76a491e853c938c421694a29456975fe}, +volume = {40}, +year = {2000} +} +@book{Lindell1995, +author = {Lindell, IV Ismo V}, +booktitle = {Electromagnetics}, +editor = {Press, IEEE}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Ebooks/Electromagnetisme/Lindell/Lindell{\_}MethodsForElectromagneticAnalysis.pdf:pdf}, +isbn = {978-0198562399}, +title = {{Methods for Electromagnetic Field Analysis}}, +url = {http://www.osti.gov/energycitations/product.biblio.jsp?osti{\_}id=6588267}, +year = {1995} } @article{Stix1965, abstract = {... 2. Same as Fig. 1, except that it is assumed that kz2c2/Wpi2 {\textgreater} 2, in accordance with the criterion IT. H. Stix, The Theory of Plasma Waves (McGraw- Hill Book Company, Inc., New York, 1962), p. 651 for accessibility to the lower hybrid resonance. ...}, author = {Stix, Thomas H.}, doi = {10.1103/PhysRevLett.15.878}, -file = {::}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Stix1965{\_}Radiation and Absorption Via Mode Conversion in an Inhomogeneous Collision-Free Plasma.pdf:pdf}, isbn = {0031-9007}, issn = {00319007}, journal = {Phys. Rev. Lett.}, @@ -469,107 +444,57 @@ @article{Stix1965 volume = {15}, year = {1965} } -@article{Guilhem2011, -abstract = {A new concept of multijunction-type antenna has been developed, the Passive-Active Multijunction, which improves the cooling of the waveguides and the damping of the neutron energy (for ITER) compared to Full Active Multijunction. Due to the complexity of the structures, prototypes of the mode converters and of the Passive-Active-Multijunction launcher were fabricated and tested, in order to validate the different manufacturing processes and the manufacturer's capability to face this challenging project. This paper describes the manufacturing process, the tests of the various prototypes and the construction of the final Passive-Active-Multijunction launcher, which entered into operation in October 2009. It has been commissioned and is fully operational on the Tore-Supra tokamak, since design objectives were reached in March 2010: 2.75 MW - 78 s, power density of 25 MW/m2 in active waveguides, steady-state apparent surface temperatures {\textless}350 °C; 10 cm long distance coupling. {\textcopyright} 2011 Elsevier B.V. All rights reserved.}, -author = {Guilhem, D. and Samaille, F. and Bertrand, Bernard and Lipa, M. and Achard, Joelle and Agarici, Gilbert and Argouarch, A. and Armitano, A. and Belo, J.H. H. and Bej, Z. and Berger-By, G. and Bouquey, F. and Brun, C. and Chantant, M. and E.corbel and Delmas, E. and Delpech, L{\'{e}}na and Doceul, L. and Ekedahl, Annika and Faisse, F. and Fejoz, P. and Goletto, C. and Goniche, M. and Hatchressian, J.C. C. and Hillairet, J. and Houry, M. and Joanard, J.P. P. and Joubert, P. and Lambert, R. and Lombard, G. and Lyonne, M. and Madeleine, S. and Magne, R. and Marfisi, L. and Martinez, A. and Maury, M. and Missirlian, M. and Mollard, Patrick and Poli, S. and Portafaix, C. and Preynas, M{\'{e}}lanie and Prou, Marc and Raulin, D. and Rousset, E. and Saille, A. and Soler, B. and Thouvenin, D. and Verger, Jean-Marc and Volpe, D. and Vulliez, Karl and Zago, B. and Corbel, E.}, -doi = {10.1016/j.fusengdes.2011.02.006}, -file = {:home/hash/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Guilhem et al. - 2011 - Manufacturing process and tests of a lower hybrid passive active multi-junction launcher for long pulse experime.pdf:pdf}, -issn = {09203796}, -journal = {Fusion Engineering and Design}, -keywords = {ITER,Long pulse,Lower Hybrid,Lower hybrid,Passive-,Passive-active,RF h,[Brazing}, -number = {6-8}, -pages = {6--10}, -publisher = {Elsevier B.V.}, -title = {{Manufacturing process and tests of a Lower Hybrid Passive-Active Multijunction launcher for long pulse experiments on Tore-Supra}}, -url = {http://dx.doi.org/10.1016/j.fusengdes.2011.02.006}, -volume = {86}, +@article{Decker2011, +abstract = {A detailed study of lower hybrid current drive (LHCD) in ITER is provided, focusing on the wave propagation and current drive mechanisms. A combination of ray-tracing and Fokker?Planck calculations are presented for various plasma scenarios, wave frequency and polarization. The dependence of the driven current and the location of power deposition upon the coupled wave spectrum is systematically determined, in order to set objectives for the antenna design. The respective effects of finite-power levels, magnetic trapping, and detailed antenna spectra are accounted for and quantitatively estimated. The sensitivity of LHCD to density and temperature profiles is calculated. From the simulation results, an optimum value for the parallel index of refraction is proposed as a compromise between efficiency and robustness with respect to those profile variations. The corresponding current drive efficiency is found to be similar for the two frequencies generally considered for ITER, f = 3.7 GHz and f = 5.0 GHz.}, +author = {Decker, Joan and Peysson, Yves and Hillairet, J. and Artaud, J.-F. and Basiuk, V. and Becoulet, A. and Ekedahl, Annika and Goniche, M. and Hoang, G T and Imbeaux, F. and Ram, A K and Schneider, M.}, +doi = {10.1088/0029-5515/51/7/073025}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Decker2011.pdf:pdf;:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Decker2011{\_}Calculations of Lower Hybrid Current Drive in ITER.pdf:pdf}, +issn = {0029-5515}, +journal = {Nuclear Fusion}, +number = {7}, +pages = {73025}, +title = {{Calculations of lower hybrid current drive in ITER}}, +url = {http://stacks.iop.org/0029-5515/51/i=7/a=073025}, +volume = {51}, year = {2011} } -@techreport{Moreau1984, -author = {Moreau, Didier and Nguyen, T K}, -institution = {{\{}EUR-CEA-FC1246{\}} Euratom-{\{}CEA{\}}}, -title = {{Couplage de l'onde lente au voisinage de la fr{\'{e}}quence hybride basse dans les grands Tokamaks}}, -year = {1984} -} -@techreport{Koert2008a, -author = {Koert, P and MacGibbon, P and Vieira, R and Terry, D and Leccacorvi, R and Doody, J and Beck, W}, -file = {::}, -institution = {PSFC/JA-08-50}, -title = {{Waveguide Splitter for Lower Hybrid Current Drive}}, -url = {http://www.new.ans.org/pubs/journals/fst/a{\_}8885}, -year = {2008} -} -@article{Krapchev1978, -abstract = {The launching of RF waves from a two- and four-waveguide array is studied. From Brambilla's grill theory, analytic expressions for the reflection coefficients, the power spectrum and the fields in the slab model of a cold inhomogeneous plasma are derived. As a first approximation to the coupling problem, only the fundamental mode in the waveguides is considered, the fast wave in the plasma is neglected and the density profile near the plasma edge is assumed to be linear. It is shown that for a fourwaveguide array with alternate 0, ? phases, optimum coupling is obtained when the inside waveguides carry about four times the power of the outside ones. This reduces and equalizes the waveguide reflection coefficients and avoids excessive shifts of the penetrated power spectrum to low values of the wavenumber parallel to the magnetic field.}, -author = {Krapchev, Vladimir B. and Bers, Abraham}, -doi = {10.1088/0029-5515/18/4/008}, -file = {::;::}, +@article{Gormezano1985, +abstract = {A new multijunction grill-type launcher has been tested on the Petula-B tokamak. In this new launcher, the RF power is divided by means of an E-plane junction, and the phase between each resultant wave is obtained by a suitable reduction in the height of the waveguides. Data obtained on Petula-B indicate that both the heating efficiency (4.5 eV � 10 13 cm ?3 �kW ?1 ) and the parametric dependences of the reflection coefficient are very similar to those of a conventional grill. Therefore, such a multijunction grill may greatly simplify the construction of grills considered for use in future large-scale tokamaks.}, +author = {Gormezano, C. and Briand, P and Briffod, G and Hoang, G.T. T and N'Guyen, T.K. K and Moreau, D and Ray, G and Search, Home and Journals, Collections and Contact, About and Iopscience, My and Address, I P}, +doi = {10.1088/0029-5515/25/4/002}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Gormezano1985{\_}Lower-hybrid plasma heating via a new launcher – the multijunction grill.pdf:pdf}, issn = {0029-5515}, journal = {Nuclear Fusion}, +month = {apr}, number = {4}, -pages = {519}, -title = {{Waveguide array excitation of lower hybrid fields in a tokamak plasma}}, -url = {http://stacks.iop.org/0029-5515/18/i=4/a=008}, -volume = {18}, -year = {1978} -} -@article{Ehst1982, -annote = {From Duplicate 1 (Lower hybrid heating and current drive system for a tokamak reactor - Ehst, David A; Boley, Charles D; Evans, Kenneth; Jung, Jungchung; Trachsel, Clarence A; Hino, Tomoaki) - -From Duplicate 1 (Lower hybrid heating and current drive system for a tokamak reactor - Ehst, David A; Boley, Charles D; Evans, Kenneth; Jung, Jungchung; Trachsel, Clarence A; Hino, Tomoaki) - -10.1007/BF01052392}, -author = {Ehst, David A and Boley, Charles D and Evans, Kenneth and Jung, Jungchung and Trachsel, Clarence A and Hino, Tomoaki}, -file = {::}, -issn = {0164-0313}, -journal = {Journal of Fusion Energy}, -keywords = {current drive,lower hybrid heating,steady-state tokamak,tokamak reactor}, -number = {1}, -pages = {83--109}, -publisher = {Springer Netherlands}, -title = {{Lower hybrid heating and current drive system for a tokamak reactor}}, -url = {http://dx.doi.org/10.1007/BF01052392}, -volume = {2}, -year = {1982} -} -@misc{Bradley2007, -author = {Bradley, Scott}, -title = {{Sign Conventions in Electromagnetic (EM) Waves}}, -url = {https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-007-electromagnetic-energy-from-motors-to-lasers-spring-2011/readings/MIT6{\_}007S11{\_}sign.pdf}, -year = {2007} +pages = {419--423}, +title = {{Lower-hybrid plasma heating via a new launcher – the multijunction grill}}, +url = {http://stacks.iop.org/0029-5515/25/i=4/a=002 http://stacks.iop.org/0029-5515/25/i=4/a=002?key=crossref.8b1a32f0951c4359e809652554529150}, +volume = {25}, +year = {1985} } -@article{Bernabei1975, -author = {Bernabei, S. and Heald, M A and Hooke, W M and Paoloni, F J}, -doi = {10.1103/PhysRevLett.34.866}, -file = {::}, -journal = {Phys. Rev. Lett.}, -number = {14}, -pages = {866--870}, -publisher = {American Physical Society}, -title = {{Penetration of Slow Waves into a Dense Plasma Using a Phased Wave-Guide Array}}, -url = {http://link.aps.org/doi/10.1103/PhysRevLett.34.866}, -volume = {34}, -year = {1975} +@book{Lindell2004, +abstract = {The presentation in this book is a true reflection of the author's grasp of the subject and his skills as a writer.}, +author = {Lindell, Ismo V}, +booktitle = {Book}, +doi = {10.1002/0471723096}, +isbn = {978-0471648017}, +keywords = {Science}, +pages = {253}, +title = {{Differential Forms in Electromagnetics}}, +year = {2004} } -@article{Brambilla1979, -abstract = {... The theory of waveguide launching of lower hybrid (LH) waves has been developed in Refs [1-5] and found to be in satisfactory agreement with ... the fraction of power able to penetrate into the plasma core essentially depends on the value of the critical index for accessibility , Eq.(l ...}, -author = {Brambilla, Marco}, -doi = {10.1088/0029-5515/19/10/006}, -file = {::}, -issn = {0029-5515}, -journal = {Nuclear Fusion}, -month = {oct}, -number = {10}, -pages = {1343--1357}, -title = {{Waveguide launching of lower hybrid waves}}, -url = {http://stacks.iop.org/0029-5515/19/i=10/a=006?key=crossref.f75bd0add10137b71a4d878a71971b22}, -volume = {19}, -year = {1979} +@book{Swanson2003, +author = {Swanson, D Gary}, +isbn = {0 7503 0927 X}, +publisher = {Taylor {\&} Francis, 2nd edition, Bristol}, +title = {{Plasma Waves}}, +year = {2003} } @article{Motley1980B, author = {Motley, R.W. and Bernabei, S. and Hooke, W.M. and Paoloni, F.J.}, doi = {10.1088/0029-5515/20/10/002}, -file = {::}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Theses/Motley1980{\_}PhasedWaveguideArrayWithFixedTuningElements.pdf:pdf}, issn = {0029-5515}, journal = {Nuclear Fusion}, month = {oct}, @@ -580,135 +505,115 @@ @article{Motley1980B volume = {20}, year = {1980} } -@article{Warnick2014, -author = {Warnick, Karl F. and Russer, Peter H.}, -doi = {10.2528/PIER14063009}, -issn = {1559-8985}, -journal = {Progress In Electromagnetics Research}, -pages = {83--112}, -title = {{DIFFERENTIAL FORMS AND ELECTROMAGNETIC FIELD THEORY}}, -url = {http://www.jpier.org/PIER/pier.php?paper=14063009}, -volume = {148}, -year = {2014} -} -@article{Mirizzi2003, -abstract = {This paper outlines the preliminary radiofrequency analysis of the prominent microwave components of the LHCD system for ITER. The general overview and inclusive analysis of the system is given in Ph. Bibet et al. (Overview of the ITER-Feat LH System; this Conference), a companion paper in this Conference. The results of the analysis and optimisation of the most relevant components is reported as computed by the 'high frequency structure simulator' (HFSS), a computer code developed by ANSOFT and based on the finite elements method. ?? 2003 Elsevier Science B.V. All rights reserved.}, -author = {Mirizzi, F. and Bibet, Philippe and Kuzikov, S.}, -doi = {10.1016/S0920-3796(03)00081-4}, -file = {::}, -isbn = {0920-3796}, -issn = {09203796}, -journal = {Fusion Engineering and Design}, -keywords = {Components,Microwave,Radiofrequency}, -pages = {487--490}, -title = {{The main microwave components of the LHCD system for ITER}}, -volume = {66-68}, -year = {2003} +@article{Golant1972, +author = {Golant, V. E.}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Golant1971.pdf:pdf;:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Petelin{\_}Kasparek1991.pdf:pdf}, +journal = {Soviet Physics - Technical Physics}, +number = {12}, +pages = {1980--1988}, +title = {{Plasma Penetration Near The Lower Hybrid Frequency}}, +url = {papers2://publication/uuid/C77B2967-1EA5-4BC7-9523-8505AF8B295D}, +volume = {16}, +year = {1972} } -@article{Briggs1972, -author = {Briggs, R. J. and Parker, Ronald R}, -doi = {10.1103/PhysRevLett.29.852}, -file = {::}, +@article{Porkolab1984, +author = {Porkolab, M and Schuss, J J and Lloyd, B and Takase, Y and Texter, S and Bonoli, P. and Fiore, C and Gandy, R and Gwinn, D and Lipschultz, B and Marmar, E and Pappas, D and Parker, R and Pribyl, P}, +doi = {10.1103/PhysRevLett.53.450}, issn = {0031-9007}, -journal = {Phys. Rev. Lett.}, -month = {sep}, -number = {13}, -pages = {852--855}, +journal = {Physical Review Letters}, +month = {jul}, +number = {5}, +pages = {450--453}, publisher = {American Physical Society}, -title = {{Transport of rf Energy to the Lower Hybrid Resonance in an Inhomogeneous Plasma}}, -url = {http://link.aps.org/doi/10.1103/PhysRevLett.29.852}, -volume = {29}, -year = {1972} +title = {{Observation of Lower-Hybrid Current Drive at High Densities in the Alcator {\textless}math display="inline"{\textgreater} {\textless}mi{\textgreater}C{\textless}/mi{\textgreater} {\textless}/math{\textgreater} Tokamak}}, +url = {http://link.aps.org/doi/10.1103/PhysRevLett.53.450}, +volume = {53}, +year = {1984} } -@article{Karney1978a, -abstract = {The motion of an ion in a lower hybrid wave in a tokamak type plasma is studied. For ions with v⊥ ≥ $\omega$/k⊥, the motion is stochastic for fields satisfying E/B0 {\textgreater} ¼($\Omega$i /$\omega$)1/3($\omega$/k⊥). Provided that the perpendicular phase velocity, $\omega$/k⊥, can be slowed down to a few times the ion thermal speed, this stochastic ion motion may be an important mechanism by which injected rf power near the lower hybrid frequency can directly heat the ions.}, -archivePrefix = {arXiv}, -arxivId = {physics/0501034}, -author = {Karney, Charles F. F.}, -doi = {10.1063/1.862406}, -eprint = {0501034}, -file = {::}, -issn = {00319171}, -journal = {Physics of Fluids}, -keywords = {IONS,NONLINEAR PROBLEMS,PLASMA HEATING,PLASMA W}, -number = {9}, -pages = {1584}, -primaryClass = {physics}, -publisher = {AIP}, -title = {{Stochastic ion heating by a lower hybrid wave}}, -url = {https://www.zenodo.org/record/32013?ln=en http://scitation.aip.org/content/aip/journal/pof1/21/9/10.1063/1.862406}, -volume = {21}, -year = {1978} +@techreport{Bers1981, +author = {Bers, A and Theilhaber, K S}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Bers1981.pdf:pdf}, +institution = {Plasma Fusion Center, MIT, Cambridge, MA}, +number = {PFC/JA-81-20}, +title = {{Linear Theory of Coupling EM Power from Waveguide Arrays to a Plasma in a Magnetic Field}}, +year = {1981} } -@book{Benford2015, -author = {Benford, James and Swegle, John A and Schamiluglu, Edl}, -edition = {third edit}, -isbn = {9781482260595}, -keywords = {Multipactor}, -mendeley-tags = {Multipactor}, -pages = {453}, -publisher = {CRC Press}, -title = {{High Power Microwaves}}, -year = {2015} +@article{Bonoli1982, +author = {Bonoli, Paul T.}, +doi = {10.1063/1.863744}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Bonoli1982{\_}Toroidal and scattering effects on lower-hybrid wave propagation.pdf:pdf}, +issn = {00319171}, +journal = {Physics of Fluids}, +number = {2}, +pages = {359}, +title = {{Toroidal and scattering effects on lower-hybrid wave propagation}}, +url = {http://scitation.aip.org/content/aip/journal/pof1/25/2/10.1063/1.863744}, +volume = {25}, +year = {1982} } -@article{Gormezano1985, -abstract = {A new multijunction grill-type launcher has been tested on the Petula-B tokamak. In this new launcher, the RF power is divided by means of an E-plane junction, and the phase between each resultant wave is obtained by a suitable reduction in the height of the waveguides. Data obtained on Petula-B indicate that both the heating efficiency (4.5 eV � 10 13 cm ?3 �kW ?1 ) and the parametric dependences of the reflection coefficient are very similar to those of a conventional grill. Therefore, such a multijunction grill may greatly simplify the construction of grills considered for use in future large-scale tokamaks.}, -author = {Gormezano, C. and Briand, P and Briffod, G and Hoang, G.T. T and N'Guyen, T.K. K and Moreau, D and Ray, G and Search, Home and Journals, Collections and Contact, About and Iopscience, My and Address, I P}, -doi = {10.1088/0029-5515/25/4/002}, -file = {::}, +@article{ITERPhysics_Chap6, +author = {Drive, ITER Physics Expert Group on Energe and Editors, ITER Physics Basis}, +doi = {10.1088/0029-5515/40/7/512}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/ITER physics basis{\_}Heating and Current Drive.pdf:pdf}, issn = {0029-5515}, journal = {Nuclear Fusion}, -month = {apr}, -number = {4}, -pages = {419--423}, -title = {{Lower-hybrid plasma heating via a new launcher – the multijunction grill}}, -url = {http://stacks.iop.org/0029-5515/25/i=4/a=002 http://stacks.iop.org/0029-5515/25/i=4/a=002?key=crossref.8b1a32f0951c4359e809652554529150}, -volume = {25}, -year = {1985} +month = {jul}, +number = {7}, +pages = {1429--1429}, +title = {{ITER Physics Basis Chapter 6: Plasma auxiliary heating and current drive}}, +url = {http://stacks.iop.org/0029-5515/40/i=7/a=512?key=crossref.76a491e853c938c421694a29456975fe}, +volume = {40}, +year = {2000} } -@article{Ikeda1989, -author = {Ikeda, Y and Imai, T and Ushigusa, K and Seki, M and Konishi, K and Naito, O and Honda, M and Kiyono, K and Maebara, S and Nagashima, T and Sawahata, M and Suganuma, K and Suzuki, N and Uehara, K and Yokokura, K and Team, JT-60}, -doi = {10.1088/0029-5515/29/10/016}, -file = {::;::}, -isbn = {0029-5515}, -issn = {17414326}, +@article{Ekedahl2010b, +abstract = {A new ITER-relevant lower hybrid current drive (LHCD) launcher, based on the passive-active-multijunction (PAM) concept, was brought into operation on the Tore Supra tokamak in autumn 2009. The PAM launcher concept was designed in view of ITER to allow efficient cooling of the waveguides, as required for long pulse operation. In addition, it offers low power reflection close to the cut-off density, which is very attractive for ITER, where the large distance between the plasma and the wall may bring the density in front of the launcher to low values. The first experimental campaign on Tore Supra has shown extremely encouraging results in terms of reflected power level and power handling. Power reflection coefficient {\textless}2{\%} is obtained at low density in front of the launcher, i.e. close to the cut-off density, and very good agreement between the experimental results and the coupling code predictions is obtained. Long pulse operation at ITER-relevant power density has been demonstrated. The maximum power and energy reached so far is 2.7MW during 78 s, corresponding to a power density of 25MWm-2, i.e. its design value at f = 3.7 GHz. In addition, 2.7MWhas been coupled at a plasma-launcher distance of 10 cm, with a power reflection coefficient {\textless}2{\%}. Finally, full non-inductive discharges have been sustained for 50 s with the PAM. {\textcopyright} 2010 IOP Publishing Ltd.}, +author = {Ekedahl, Annika and Delpech, L{\'{e}}na and Goniche, M. and Guilhem, D. and Hillairet, J. and Preynas, M{\'{e}}lanie and Sharma, P.K. and Achard, Joelle and Bae, Y.S. S. and Bai, X. and Balorin, C. and Baranov, Y. and Basiuk, V. and B{\'{e}}coulet, A. and Belo, J. and Berger-By, G. and Bremond, S. and Castaldo, C. and Ceccuzzi, Silvio and Cesario, Roberto and Corbel, E. and Courtois, X. and Decker, Joan and Delmas, E. and Ding, X. and Douai, D. and Goletto, C. and Gunn, J.P. and Hertout, P. and Hoang, G.T. and Imbeaux, F. and Kirov, K.K. and Litaudon, X. and Magne, R. and Mailloux, Joelle and Mazon, D. and Mirizzi, F. and Mollard, Patrick and Moreau, P. and Oosako, T. and Petr{\v{z}}{\'{i}}lka, V. A. and Peysson, Yves and Poli, S. and Prou, Marc and Saint-Laurent, F. and Samaille, F. and Saoutic, B. and Others}, +doi = {10.1088/0029-5515}, +issn = {00295515 17414326}, journal = {Nuclear Fusion}, -number = {10}, -pages = {1815--1819}, -title = {{Efficient Lower Hybrid Current Drive Using a Multijunction Launcher on Jt-60}}, -url = {http://stacks.iop.org/0029-5515/29/i=10/a=016}, -volume = {29}, -year = {1989} +number = {11}, +title = {{Validation of the ITER-relevant passive-active-multijunction LHCD launcher on long pulses in Tore Supra}}, +volume = {50}, +year = {2010} } -@article{Brambilla1976b, -abstract = {The coupling efficiency of a phased multi-waveguide structure (the "Grill") designed to launch HF-waves at the lower hybrid resonance to heat large toroidal plasmas, while satisfying the accessibility condition, is studied. To find the reflection and transmission coefficients, as well as the kn-spectrum of the excited field, the waveguide field, represented as a superposition of eigenmodes, is matched to the field in the plasma, which is evaluated on the assumption of a linear density profile near the plasma edge. It is found that the reflection coefficient can be made acceptably low and is not sensitively dependent on the plasma parameters. It is concluded that it is possible to design a Grill capable to launch lower hybrid waves at the power level required for the ignition of a reactor plasma.}, +@book{Brambilla1998, +abstract = {The book deals with the propagation and absorption of high frequency waves in plasmas. The text collects in a structured and self-contained way the basic knowledge on the broad and varied behavior of plasma waves, adopting the microscopic kinetic description of the plasma as unifying principle. The internal coherence of the theory is explicitly stressed, and interesting physical phenomena peculiar to plasmas are discussed in detail, including collisionless damping of waves, the development of stochasticity in the interactions of charged particles with electromagnetic waves, and nonlinear interactions between waves. The most common and useful approximations used in solving practical problems are derived as special cases from the more general kinetic approach, thereby clarifying their meaning and domain of applicability. This exposition should be useful to plasma physicists both as an introduction and a reference to this field of research.}, author = {Brambilla, Marco}, -doi = {10.1088/0029-5515/16/1/005}, -file = {::}, -isbn = {0029-5515}, -issn = {0029-5515}, -journal = {Nuclear Fusion}, -number = {1}, -pages = {47--54}, -title = {{Slow-wave launching at the lower hybrid frequency using a phased waveguide array}}, -volume = {16}, -year = {1976} +doi = {10.1088/0029-5515/38/11/701}, +isbn = {0198559569}, +issn = {00295515}, +pages = {656}, +publisher = {Oxford University Press}, +title = {{Kinetic Theory of Plasma Waves: Homogeneous Plasmas}}, +url = {http://www.amazon.com/Kinetic-Theory-Plasma-Waves-International/dp/0198559569{\#}}, +year = {1998} } -@article{Decker2011, -abstract = {A detailed study of lower hybrid current drive (LHCD) in ITER is provided, focusing on the wave propagation and current drive mechanisms. A combination of ray-tracing and Fokker?Planck calculations are presented for various plasma scenarios, wave frequency and polarization. The dependence of the driven current and the location of power deposition upon the coupled wave spectrum is systematically determined, in order to set objectives for the antenna design. The respective effects of finite-power levels, magnetic trapping, and detailed antenna spectra are accounted for and quantitatively estimated. The sensitivity of LHCD to density and temperature profiles is calculated. From the simulation results, an optimum value for the parallel index of refraction is proposed as a compromise between efficiency and robustness with respect to those profile variations. The corresponding current drive efficiency is found to be similar for the two frequencies generally considered for ITER, f = 3.7 GHz and f = 5.0 GHz.}, -author = {Decker, Joan and Peysson, Yves and Hillairet, J. and Artaud, J.-F. and Basiuk, V. and Becoulet, A. and Ekedahl, Annika and Goniche, M. and Hoang, G T and Imbeaux, F. and Ram, A K and Schneider, M.}, -doi = {10.1088/0029-5515/51/7/073025}, -file = {:home/hash/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Decker et al. - 2011 - Calculations of lower hybrid current drive in ITER(2).pdf:pdf;::}, +@book{Benford1992, +address = {Boston}, +author = {Benford, James and Swegle, John A}, +edition = {first edit}, +editor = {{Artech House}}, +isbn = {0890064156}, +pages = {412}, +title = {{High Power Microwaves}}, +year = {1992} +} +@article{Puri1974, +abstract = {Previously, it has been shown that in a cold, inhomogeneous, magnetized plasma half-space the lower-hybrid resonance is accessible to the transverse-magnetic (TM) plane waves incident on the vacuum-plasma interface at an approximately grazing incidence, provided that at the hybrid layer ? pe /? ce ? 0.4. In this paper, these results are extended to the slow-wave case when n z , the refractive index in the static magnetic field direction, exceeds unity. It is found that the plasma is indeed accessible to the slow waves if Golant's accessibility criterion n z {\textgreater} 1 + (? pe /? ce ) 2 is satisfied. The following recommendations can be made for coupling r.f. energy to the lower-hybrid resonance: (i) if ? pe /? ce ? 0.4, efficient coupling is possible by launching TEM-like waves on the plasma column, (ii) if ? pe /? ce ? 0.4 and if the transverse machine dimensions exceed the r.f. vacuum wavelength, it is possible to couple TM waves using passive slow-wave structures inside the machine walls, (iii) if ? pe /? ce ? 0.4, but for smaller machine dimensions, recourse must be taken to transverse-electric slow-wave coupling with current-carrying coils of appropriate periodicity . If, as was pointed out by Glagolev, propagation from the plasma edge to the hybrid layer is not materially affected by the inclusion of finite-temperature effects, by far the most elegant solution (with potential application to thermonuclear plasmas) for coupling r.f. energy from the second to the twentieth ion-cyclotron harmonic waves is by launching TEM-like waves in the coaxial waveguide formed by the plasma columnand the containing walls.}, +author = {Puri, S and Tutter, M}, +doi = {10.1088/0029-5515/14/1/014}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Puri1974{\_}Slow-wave coupling to the lower-hybrid resonance.pdf:pdf}, issn = {0029-5515}, journal = {Nuclear Fusion}, -number = {7}, -pages = {73025}, -title = {{Calculations of lower hybrid current drive in ITER}}, -url = {http://stacks.iop.org/0029-5515/51/i=7/a=073025}, -volume = {51}, -year = {2011} +month = {jan}, +number = {1}, +pages = {93--101}, +title = {{Slow-wave coupling to the lower-hybrid resonance}}, +url = {http://stacks.iop.org/0029-5515/14/i=1/a=014 http://stacks.iop.org/0029-5515/14/i=1/a=014?key=crossref.f33f933ccc2480cfbaabffe38ddbffa5}, +volume = {14}, +year = {1974} } -@inproceedings{Lallia1975, +@proceedings{Lallia1975, author = {Lallia, P.}, booktitle = {In Tex. Tech. Univ. Proc. of the 2d Topical Conf. on RF Plasma Heating 6 p}, editor = {Lallia, P}, @@ -717,71 +622,60 @@ @inproceedings{Lallia1975 url = {http://adsabs.harvard.edu/abs/1975rfph.conf.....L}, year = {1975} } -@book{Kolundzija2002, -author = {Kolund{\v{z}}ija, Branko and Djordjevi{\'{c}}, A.}, -isbn = {0890063605}, -pages = {408}, -publisher = {Artech House}, -title = {{Electromagnetic Modeling of Composite Metallic and Dielectric Structures}}, -year = {2002} -} -@inproceedings{Tonon1982, -author = {Tonon, G and Others}, -booktitle = {International Atomic Energy Agency (IAEA): IAEA.}, -title = {{Lower Hybrid Current Drive and heating on the {\{}WEGA{\}} Tokamak}}, -year = {1983} -} -@article{Brambilla1983, -abstract = {The authors have investigated quasi-linear ion Landau damping of lower hybrid waves in an inhomogeneous plasma. To this end, they have simultaneously solved Maxwell equations and the ion kinetic equation, starting from the antenna and proceeding towards the centre of a plane-layered plasma. As a consequence of the development of a suprathermal tail in the ion distribution function, the efficiency of the absorption increases and the absorption region is found to shift to lower densities as the launched power increases. Absorption is always complete at the layer where the wave phase velocity equals about three times the local ion thermal velocity, usually somewhat before the linear turning point is reached.}, -author = {Brambilla, Marco and Chen, Yan-ping}, -journal = {Nuclear Fusion}, -number = {4}, -pages = {541}, -title = {{Quasi-linear ion heating by lower hybrid waves}}, -url = {http://stacks.iop.org/0029-5515/23/i=4/a=013}, -volume = {23}, -year = {1983} +@book{Clemmow1996, +author = {Clemmow, P. C.}, +booktitle = {Vacuum}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Ebooks/Electromagnetisme/Clemmow/0198592256Wave{\_}Spectrum{\_}RepresentationB.pdf:pdf}, +isbn = {0-7803-3411-6}, +publisher = {Wiley-IEEE Press}, +title = {{The Plane Wave Spectrum Representation of Electromagnetic Fields}}, +url = {http://www.getcited.org/pub/100124136}, +year = {1996} +} +@book{Schwinger1998, +author = {Schwinger, J}, +booktitle = {Quantum fields and strings: a course for {\ldots}}, +isbn = {0813346622}, +pages = {592}, +publisher = {Avalon Publishing}, +title = {{Classical Electrodynamics}}, +year = {1998} } -@article{Porkolab1984, -author = {Porkolab, M and Schuss, J J and Lloyd, B and Takase, Y and Texter, S and Bonoli, P. and Fiore, C and Gandy, R and Gwinn, D and Lipschultz, B and Marmar, E and Pappas, D and Parker, R and Pribyl, P}, -doi = {10.1103/PhysRevLett.53.450}, -issn = {0031-9007}, -journal = {Physical Review Letters}, +@inproceedings{Hoang2009, +abstract = {A 20 MW/5 GHz lower hybrid current drive (LHCD) system was initially due to be commissioned and used for the second mission of ITER, i.e. the Q = 5 steady state target. Though not part of the currently planned procurement phase, it is now under consideration for an earlier delivery. In this paper, both physics and technology conceptual designs are reviewed. Furthermore, an appropriate work plan is also developed. This work plan for design, R{\&}D, procurement and installation of a 20 MW LHCD system on ITER follows the ITER Scientific and Technical Advisory Committee (STAC) T13-05 task instructions. It gives more details on the various scientific and technical implications of the system, without presuming on any work or procurement sharing amongst the possible ITER partnersb The LHCD system of ITER is not part of the initial cost sharing.. This document does not commit the Institutions or Domestic Agencies of the various authors in that respect. {\textcopyright} 2009 IAEA, Vienna.}, +author = {Hoang, G.T. and B{\'{e}}coulet, A. and Jacquinot, J. and Artaud, J.F. J.-F. and Bae, Y.S. and Beaumont, B. and Belo, J.H. and Berger-By, G. and Bizarro, J.P.S. Jo{\~{a}}o P.S. and Bonoli, P. and Cho, M.H. and Decker, Joan and Delpech, L{\'{e}}na and Ekedahl, Annika and Garcia, J. and Giruzzi, G. and Goniche, M. and Gormezano, C. and Guilhem, D. and Hillairet, J. and Imbeaux, F. and Kazarian, Fabienne and Kessel, C. and Kim, S.H. and Kwak, J.G. and Jeong, J.H. and Lister, J.B. and Litaudon, X. and Magne, R. and Milora, S. and Mirizzi, F. and Namkung, W. and Noterdaeme, J.M. and Park, S.I. and Parker, R. and Peysson, Yves and Rasmussen, D. and Sharma, P.K. and Schneider, M. and Synakowski, E. and Tanga, A. and Tuccillo, A. and Wan, Y.X.}, +booktitle = {Nuclear Fusion}, +doi = {10.1088/0029-5515/49/7/075001}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Hoang2008.pdf:pdf;:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Hoang2008a.pdf:pdf}, +issn = {0029-5515}, month = {jul}, -number = {5}, -pages = {450--453}, -publisher = {American Physical Society}, -title = {{Observation of Lower-Hybrid Current Drive at High Densities in the Alcator {\textless}math display="inline"{\textgreater} {\textless}mi{\textgreater}C{\textless}/mi{\textgreater} {\textless}/math{\textgreater} Tokamak}}, -url = {http://link.aps.org/doi/10.1103/PhysRevLett.53.450}, -volume = {53}, -year = {1984} -} -@book{Brambilla1998, -abstract = {The book deals with the propagation and absorption of high frequency waves in plasmas. The text collects in a structured and self-contained way the basic knowledge on the broad and varied behavior of plasma waves, adopting the microscopic kinetic description of the plasma as unifying principle. The internal coherence of the theory is explicitly stressed, and interesting physical phenomena peculiar to plasmas are discussed in detail, including collisionless damping of waves, the development of stochasticity in the interactions of charged particles with electromagnetic waves, and nonlinear interactions between waves. The most common and useful approximations used in solving practical problems are derived as special cases from the more general kinetic approach, thereby clarifying their meaning and domain of applicability. This exposition should be useful to plasma physicists both as an introduction and a reference to this field of research.}, -author = {Brambilla, Marco}, -doi = {10.1088/0029-5515/38/11/701}, -isbn = {0198559569}, -issn = {00295515}, -pages = {656}, -publisher = {Oxford University Press}, -title = {{Kinetic Theory of Plasma Waves: Homogeneous Plasmas}}, -url = {http://www.amazon.com/Kinetic-Theory-Plasma-Waves-International/dp/0198559569{\#}}, -year = {1998} +number = {7}, +title = {{A lower hybrid current drive system for ITER}}, +url = {http://stacks.iop.org/0029-5515/49/i=7/a=075001?key=crossref.b48bdcea42107a5a9be1a7fb1e48a7c8}, +volume = {49}, +year = {2008} } -@book{Benford2007a, -author = {Benford, James and Swegle, John A and Schamiluglu, Edl}, -edition = {second edi}, -editor = {CRC}, -isbn = {9780750307062}, -pages = {552}, -title = {{High Power Microwaves}}, -year = {2007} +@article{Guilhem2011, +abstract = {A new concept of multijunction-type antenna has been developed, the Passive-Active Multijunction, which improves the cooling of the waveguides and the damping of the neutron energy (for ITER) compared to Full Active Multijunction. Due to the complexity of the structures, prototypes of the mode converters and of the Passive-Active-Multijunction launcher were fabricated and tested, in order to validate the different manufacturing processes and the manufacturer's capability to face this challenging project. This paper describes the manufacturing process, the tests of the various prototypes and the construction of the final Passive-Active-Multijunction launcher, which entered into operation in October 2009. It has been commissioned and is fully operational on the Tore-Supra tokamak, since design objectives were reached in March 2010: 2.75 MW - 78 s, power density of 25 MW/m2 in active waveguides, steady-state apparent surface temperatures {\textless}350 °C; 10 cm long distance coupling. {\textcopyright} 2011 Elsevier B.V. All rights reserved.}, +author = {Guilhem, D. and Samaille, F. and Bertrand, Bernard and Lipa, M. and Achard, Joelle and Agarici, Gilbert and Argouarch, A. and Armitano, A. and Belo, J.H. H. and Bej, Z. and Berger-By, G. and Bouquey, F. and Brun, C. and Chantant, M. and E.corbel and Delmas, E. and Delpech, L{\'{e}}na and Doceul, L. and Ekedahl, Annika and Faisse, F. and Fejoz, P. and Goletto, C. and Goniche, M. and Hatchressian, J.C. C. and Hillairet, J. and Houry, M. and Joanard, J.P. P. and Joubert, P. and Lambert, R. and Lombard, G. and Lyonne, M. and Madeleine, S. and Magne, R. and Marfisi, L. and Martinez, A. and Maury, M. and Missirlian, M. and Mollard, Patrick and Poli, S. and Portafaix, C. and Preynas, M{\'{e}}lanie and Prou, Marc and Raulin, D. and Rousset, E. and Saille, A. and Soler, B. and Thouvenin, D. and Verger, Jean-Marc and Volpe, D. and Vulliez, Karl and Zago, B. and Corbel, E.}, +doi = {10.1016/j.fusengdes.2011.02.006}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Guilhem2011.pdf:pdf}, +issn = {09203796}, +journal = {Fusion Engineering and Design}, +keywords = {ITER,Long pulse,Lower Hybrid,Lower hybrid,Passive-,Passive-active,RF h,[Brazing}, +number = {6-8}, +pages = {6--10}, +publisher = {Elsevier B.V.}, +title = {{Manufacturing process and tests of a Lower Hybrid Passive-Active Multijunction launcher for long pulse experiments on Tore-Supra}}, +url = {http://dx.doi.org/10.1016/j.fusengdes.2011.02.006}, +volume = {86}, +year = {2011} } @article{Bibet2000, abstract = {A new actively cooled advanced launcher is being built for Tore Supra LHCD to inject 4 MW during 1000 s at 3.7 GHz, at a power density of 25 MW/m2 (a conservative value observed in Tore Supra experiments). It is made from an array of 6×48 active and 6×9 passive waveguides. The design uses technologies which are relevant for a next step machine such that it can: (i) withstand a plasma radiated flux of 0.15 MW/m2; (ii) radiate power with spectra having peak N∥ values of 2.02±0.35; (iii) withstand a total torque of 8.6 104 N m during disruptions; (iv) allow an antenna 20 cm radial stroke adjustable in real time, (v) withstand a convected power flux of 10 MW/m2 on its guard limiter made of CFC tiles. A prototype of each new component of this antenna has been tested successfully at the nominal power with a pulse length of 1000 s.}, author = {Bibet, Philippe and Agarici, Gilbert and Chantant, M. and Cordier, J. J. and Deck, C. and Doceul, L. and Durocher, A. and Ekedahl, Annika and Froissard, Ph. and Garguiolo, L. and Garampon, L. and Goniche, M. and Hertout, P. and Kazarian, Fabienne and Lafon, D. and Portafaix, C. and Rey, G. and Samaille, F. and Surle, F. and Tonon, G.}, doi = {10.1016/S0920-3796(00)00450-6}, -file = {::}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Bibet2000.pdf:pdf}, isbn = {0920-3796}, issn = {0920-3796}, journal = {Fusion Engineering and Design}, @@ -792,22 +686,129 @@ @article{Bibet2000 volume = {51-52}, year = {2000} } -@book{Lindell2004, -abstract = {The presentation in this book is a true reflection of the author's grasp of the subject and his skills as a writer.}, -author = {Lindell, Ismo V}, -booktitle = {Book}, -doi = {10.1002/0471723096}, -isbn = {978-0471648017}, -keywords = {Science}, -pages = {253}, -title = {{Differential Forms in Electromagnetics}}, -year = {2004} +@article{Tonon1984, +author = {Tonon, G}, +doi = {10.1088/0741-3335/26/1A/313}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Tonon1984{\_}Heating and Current Drive by LH Waves On Toroidal Plasmas - Problems and perspectives.pdf:pdf}, +issn = {0741-3335}, +journal = {Plasma Physics and Controlled Fusion}, +month = {jan}, +number = {1A}, +pages = {145--155}, +title = {{Heating and current drive by LH-Waves on toroidal plasmas: problems and perspectives}}, +url = {http://stacks.iop.org/0741-3335/26/i=1A/a=313?key=crossref.48a2b98169970f4f579f24dc8c805348 http://stacks.iop.org/0741-3335/26/i=1A/a=313}, +volume = {26}, +year = {1984} +} +@article{Brambilla1976b, +abstract = {The coupling efficiency of a phased multi-waveguide structure (the "Grill") designed to launch HF-waves at the lower hybrid resonance to heat large toroidal plasmas, while satisfying the accessibility condition, is studied. To find the reflection and transmission coefficients, as well as the kn-spectrum of the excited field, the waveguide field, represented as a superposition of eigenmodes, is matched to the field in the plasma, which is evaluated on the assumption of a linear density profile near the plasma edge. It is found that the reflection coefficient can be made acceptably low and is not sensitively dependent on the plasma parameters. It is concluded that it is possible to design a Grill capable to launch lower hybrid waves at the power level required for the ignition of a reactor plasma.}, +author = {Brambilla, Marco}, +doi = {10.1088/0029-5515/16/1/005}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Brambilla1976a.pdf:pdf}, +isbn = {0029-5515}, +issn = {0029-5515}, +journal = {Nuclear Fusion}, +number = {1}, +pages = {47--54}, +title = {{Slow-wave launching at the lower hybrid frequency using a phased waveguide array}}, +volume = {16}, +year = {1976} +} +@article{Ikeda1989, +author = {Ikeda, Y and Imai, T and Ushigusa, K and Seki, M and Konishi, K and Naito, O and Honda, M and Kiyono, K and Maebara, S and Nagashima, T and Sawahata, M and Suganuma, K and Suzuki, N and Uehara, K and Yokokura, K and Team, JT-60}, +doi = {10.1088/0029-5515/29/10/016}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Ikeda1989{\_}Efficient lower hybrid current drive using a multijunction launcher on JT-60.pdf:pdf;:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Ikeda1989{\_}JT60U{\_}LH.pdf:pdf}, +isbn = {0029-5515}, +issn = {17414326}, +journal = {Nuclear Fusion}, +number = {10}, +pages = {1815--1819}, +title = {{Efficient Lower Hybrid Current Drive Using a Multijunction Launcher on Jt-60}}, +url = {http://stacks.iop.org/0029-5515/29/i=10/a=016}, +volume = {29}, +year = {1989} +} +@article{Bers1983, +author = {Bers, Abraham and Theilhaber, K S}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Bers1983.pdf:pdf}, +journal = {Nuclear Fusion}, +number = {1}, +pages = {41--48}, +title = {{Three-dimensional theory of waveguide-plasma coupling}}, +volume = {41}, +year = {1983} +} +@article{Jobes1985, +author = {Jobes, F. C. and Bernabei, S. and Chu, T. K. and Hooke, W. M. and Meservey, E. B. and Motley, R. W. and Stevens, J. E. and von Goeler, S.}, +doi = {10.1103/PhysRevLett.55.1295}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Jobes1985{\_}Current Rampup by LH Waves in the PLT Tokamak.pdf:pdf}, +issn = {0031-9007}, +journal = {Physical Review Letters}, +month = {sep}, +number = {12}, +pages = {1295--1298}, +title = {{Current Rampup by Lower-Hybrid Waves in the PLT Tokamak}}, +url = {http://link.aps.org/doi/10.1103/PhysRevLett.55.1295}, +volume = {55}, +year = {1985} +} +@misc{Michelsen2017, +author = {Michelsen, Eric}, +title = {{Funky Electromagnetic Concepts}}, +url = {physics.ucsd.edu/{~}emichels}, +year = {2017} +} +@book{Pond2008, +author = {Pond, N H}, +isbn = {9780981692302}, +publisher = {Russ Cochran, Publisher}, +title = {{The tube guys}}, +url = {http://books.google.fr/books?id=fXEfAQAAIAAJ}, +year = {2008} +} +@article{Krapchev1981, +author = {Krapchev, V B and Theilhaber, K S and Ko, K C and Bers, A}, +doi = {10.1103/PhysRevLett.46.1398}, +issn = {0031-9007}, +journal = {Physical Review Letters}, +month = {may}, +number = {21}, +pages = {1398--1401}, +publisher = {American Physical Society}, +title = {{Nonlinear Coupling of Lower-Hybrid Waves at the Edge of Tokamak Plasmas}}, +url = {http://link.aps.org/doi/10.1103/PhysRevLett.46.1398}, +volume = {46}, +year = {1981} +} +@book{Mackay2010, +abstract = {The topics of anisotropy and bianisotropy are fundamental to electromagnetics from both theoretical and experimental perspectives. These properties underpin a host of complex and exotic electromagnetic phenomena in naturally occurring materials and in relativistic scenarios, as well as in artificially produced metamaterials. As a unique guide to this rapidly developing field, the book provides a unified presentation of key classic and recent results on the studies of constitutive relations, spacetime symmetries, planewave propagation, dyadic Green functions, and homogenization of composite materials. This book also offers an up-to-date extension to standard treatments of crystal optics with coverage on both linear and weakly nonlinear regimes.}, +author = {Mackay, Tom G and {Akhlesh Lakhtakia}}, +doi = {10.1111/ddi.12150}, +isbn = {9789814289610}, +pages = {1461--1467}, +title = {{Electromagnetic Anisotropy and Bianisotropy}}, +url = {http://www.worldscientific.com/doi/suppl/10.1142/7515/suppl{\_}file/7515{\_}chap01.pdf}, +year = {2010} +} +@article{Krapchev1978, +abstract = {The launching of RF waves from a two- and four-waveguide array is studied. From Brambilla's grill theory, analytic expressions for the reflection coefficients, the power spectrum and the fields in the slab model of a cold inhomogeneous plasma are derived. As a first approximation to the coupling problem, only the fundamental mode in the waveguides is considered, the fast wave in the plasma is neglected and the density profile near the plasma edge is assumed to be linear. It is shown that for a fourwaveguide array with alternate 0, ? phases, optimum coupling is obtained when the inside waveguides carry about four times the power of the outside ones. This reduces and equalizes the waveguide reflection coefficients and avoids excessive shifts of the penetrated power spectrum to low values of the wavenumber parallel to the magnetic field.}, +author = {Krapchev, Vladimir B. and Bers, Abraham}, +doi = {10.1088/0029-5515/18/4/008}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Krapchev1978{\_}Waveguide Array Excitation Of LH Fields In Tokamak Plasma.pdf:pdf;:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Krapchev1978{\_}WaveguideArrayExcitationLH.pdf:pdf}, +issn = {0029-5515}, +journal = {Nuclear Fusion}, +number = {4}, +pages = {519}, +title = {{Waveguide array excitation of lower hybrid fields in a tokamak plasma}}, +url = {http://stacks.iop.org/0029-5515/18/i=4/a=008}, +volume = {18}, +year = {1978} } @article{Fisch1978, abstract = {Continuous toroidal electron currents, which sustain the poloidal magnetic field in tokamaks, may be generated by injecting waves with net parallel momentum into the plasma via phased waveguide arrays. Waves with high phase velocity can produce a current capable of confining a reactor plasma so that steady-state tokamak operation with acceptable power dissipation becomes possible.}, author = {Fisch, Nathaniel J.}, doi = {10.1103/PhysRevLett.41.873}, -file = {::}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Fisch1978{\_}Confining{\_}TokamakPlasma{\_}with{\_}RFCurrentDrive.pdf:pdf}, issn = {0031-9007}, journal = {Physical Review Letters}, month = {sep}, @@ -819,46 +820,24 @@ @article{Fisch1978 volume = {41}, year = {1978} } -@inproceedings{Guilhem2009, -abstract = {The design and the fabrication of a new Lower Hybrid (LH) actively cooled antenna based on the passive active concept is a part of the CIMES project (Components for the Injection of Mater and Energy in Steady-state). The major objectives of Tore-Supra program is to achieve 1000 s pulses with this LH launcher, by coupling routinely {\textgreater}3 MW of LH wave at 3.7 GHz to the plasma with a parallel index n∥=1.7±0.2. The launcher is on its way to achieve its validation tests - low power Radio Frequency (RF) measurements, vacuum and hydraulic leak tests - and will be installed and commissioned on plasma during the fall of 2009. {\textcopyright} 2009 American Institute of Physics.}, -author = {Guilhem, D. and Others and Achard, Joelle and Belo, J. and Bertrand, Bernard and Bej, Z. and Bibet, Philippe and Brun, C. and Chantant, M. and Delmas, E. and Delpech, L{\'{e}}na and Doceul, Y. and Ekedahl, Annika and Goletto, C. and Goniche, M. and Hatchressian, J. C. and Hillairet, J. and Houry, M. and Joubert, P. and Lipa, M. and Madeleine, S. and Martinez, A. and Missirlian, M. and Poli, S. and Portafaix, C. and Raulin, D. and Saille, A. and Soler, B. and Thouvenin, D. and Verger, Jean-Marc and Vulliez, Karl and Zago, B. and Bobkov, Volodymyr and Noterdaeme, Jean-Marie}, -booktitle = {Radio Frequency Power In Plasmas: Proceedings Of The 18th Topical Conference, Aip Conference Proceedings}, -doi = {10.1063/1.3273786}, -isbn = {9780735407534}, -issn = {0094243X 15517616}, -keywords = {ITER,Lower hybrid,Passive-active,Tore,[Brazing}, -pages = {435--438}, -title = {{Passive Active Multi-Junction 3.7 {\{}GHz{\}} Launcher for Tore-Supra Long Pulse Experiments : Manufacturing Process and Tests}}, -url = {http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.3273786}, -volume = {1187}, -year = {2009} -} -@article{Bonoli1982, -author = {Bonoli, Paul T.}, -doi = {10.1063/1.863744}, -file = {::}, -issn = {00319171}, -journal = {Physics of Fluids}, -number = {2}, -pages = {359}, -title = {{Toroidal and scattering effects on lower-hybrid wave propagation}}, -url = {http://scitation.aip.org/content/aip/journal/pof1/25/2/10.1063/1.863744}, -volume = {25}, -year = {1982} +@misc{Bradley2007, +author = {Bradley, Scott}, +title = {{Sign Conventions in Electromagnetic (EM) Waves}}, +url = {https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-007-electromagnetic-energy-from-motors-to-lasers-spring-2011/readings/MIT6{\_}007S11{\_}sign.pdf}, +year = {2007} } -@article{Schuss1981, -author = {Schuss, J.J. and Porkolab, M. and Takase, Y. and Cope, D. and Fairfax, S. and Greenwald, M. and Gwinn, D. and Hutchinson, I.H. and Kusse, B. and Marmar, E. and Overskei, D. and Pappas, D. and Parker, R.R. and Scaturro, L. and West, J. and Wolfe, S.}, -doi = {10.1088/0029-5515/21/4/001}, -file = {::}, -issn = {0029-5515}, -journal = {Nuclear Fusion}, -month = {apr}, -number = {4}, -pages = {427--452}, -title = {{Lower hybrid heating in the Alcator A tokamak}}, -url = {https://www.google.com/url?sa=t{\&}rct=j{\&}q={\&}esrc=s{\&}source=web{\&}cd=1{\&}cad=rja{\&}uact=8{\&}ved=0ahUKEwjg-66-19zOAhXCORQKHbmzC1kQFggcMAA{\&}url=http{\%}3A{\%}2F{\%}2Fdspace.mit.edu{\%}2Fbitstream{\%}2Fhandle{\%}2F1721.1{\%}2F94663{\%}2F80ja011{\_}full.pdf{\%}3Fsequence{\%}3D1{\&}usg=AFQjCNGhibBkLEgGPQ3RWNH}, -volume = {21}, -year = {1981} +@article{Vladimira, +author = {Vladimir, Krapchev and Krapchev, Vladimir B.}, +doi = {10.1103/PhysRevLett.42.497}, +issn = {0031-9007}, +journal = {Physical Review Letters}, +month = {feb}, +number = {8}, +pages = {497--500}, +title = {{Kinetic Thoery of the Pondermotive Effects in Plasma}}, +url = {https://link.aps.org/doi/10.1103/PhysRevLett.42.497}, +volume = {42}, +year = {1979} } @article{Karney1979, author = {Karney, Charles F. F. and Fisch, Nathaniel J.}, @@ -874,56 +853,95 @@ @article{Karney1979 volume = {22}, year = {1979} } -@article{Bers1983, -author = {Bers, Abraham and Theilhaber, K S}, -file = {::}, -journal = {Nuclear Fusion}, +@techreport{Theilhaber1979, +author = {Theilhaber, K and Bers, A}, +doi = {10.1088/0029-5515/20/5/003}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Bers1979.pdf:pdf}, +institution = {Plasma Fusion Center, MIT / JA-79-15}, +isbn = {0029-5515}, +issn = {17414326}, +title = {{Coupling to the Fast Wave at Lower Hybrid Frequencies}}, +year = {1979} +} +@inproceedings{Meneghini2010b, +author = {Meneghini, Orso and Shiraiwa, Syunichi and Johnson, David and Faust, I.C. and Kanojia, Atma and Parker, Ronald and Terry, David and Vieira, Rui and Wallace, Gregory M and {Wilson Randy; Wukitch}, Steve}, +booktitle = {American Physical Society, 52nd Annual Meeting of the APS Division of Plasma Physics}, +month = {nov}, +title = {{Coupling of LH waves with four-way-splitter antenna on Alcator C-Mod}}, +year = {2010} +} +@article{Ehst1982, +annote = {From Duplicate 1 (Lower hybrid heating and current drive system for a tokamak reactor - Ehst, David A; Boley, Charles D; Evans, Kenneth; Jung, Jungchung; Trachsel, Clarence A; Hino, Tomoaki) + +From Duplicate 1 (Lower hybrid heating and current drive system for a tokamak reactor - Ehst, David A; Boley, Charles D; Evans, Kenneth; Jung, Jungchung; Trachsel, Clarence A; Hino, Tomoaki) + +10.1007/BF01052392}, +author = {Ehst, David A and Boley, Charles D and Evans, Kenneth and Jung, Jungchung and Trachsel, Clarence A and Hino, Tomoaki}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Ehst1982.pdf:pdf}, +issn = {0164-0313}, +journal = {Journal of Fusion Energy}, +keywords = {current drive,lower hybrid heating,steady-state tokamak,tokamak reactor}, number = {1}, -pages = {41--48}, -title = {{Three-dimensional theory of waveguide-plasma coupling}}, -volume = {41}, -year = {1983} +pages = {83--109}, +publisher = {Springer Netherlands}, +title = {{Lower hybrid heating and current drive system for a tokamak reactor}}, +url = {http://dx.doi.org/10.1007/BF01052392}, +volume = {2}, +year = {1982} } -@article{Gormezano1986a, -author = {Gormezano, C}, -doi = {10.1088/0741-3335/28/9A/014}, -file = {::}, -issn = {0741-3335}, -journal = {Plasma Physics and Controlled Fusion}, +@article{Ridolfini2005, +abstract = {A prototype passive active multijunction (PAM) launcher for lower hybrid (LH) waves conceptually similar to that foreseen for ITER has been successfully tested on FTU at frequency f = 8 GHz. The power routinely and safely managed for the maximum time allowed by the LH power plant (0.9 s) without any fault in the transmission lines is 250 kW, corresponding to 75 MW m ?2 across the antenna active area and very close to the design value of 270 kW or 80 MW m ?2 . The achieved value is at least 1.4 times larger than the ITER request, which would be only 52 MW m ?2 , if the 33 MW m ?2 required to the ITER grill in order to couple 20 MW to the plasma, are scaled up linearly with f , from f ITER = 5 GHz. This linear scaling of the power handling capability of the LH antennae is indeed conservative with respect to the available data. The test results validate also the other two main expectations relevant to ITER, foreseen by the codes, namely to operate with the grill entirely in the vessel shadow and to still preserve good current drive (CD) efficiency. Even with the grill mouth retracted 2 mm inside the port shadow and with density in front of the launcher very close or even lower than the cut-off value, the PAM reflection coefficient is always ? 2.5{\%}, if the antenna has been properly conditioned. The CD efficiency is comparable to that of a conventional grill, once the lower directivity is taken into account. Flexibility in determining the N || spectrum is also maintained, according to hard x-rays and electron cyclotron emission spectra. Conditioning the PAM in order to operate at the ITER equivalent power level required only one day of radio-frequency operation, without a previous baking of the waveguides.}, +author = {Ridolfini, V Pericoli and Bibet, Philippe and Mirizzi, F and Apicella, M.L L and Barbato, E and Buratti, P. and Calabr{\`{o}}, G and Cardinali, A and Granucci, G and Panaccione, L and Podda, S and Sozzi, C and Tuccillo, A.A A and Calabr�, G and Cardinali, A and Granucci, G and Panaccione, L and Podda, S and Sozzi, C and Tuccillo, A.A A}, +doi = {10.1088/0029-5515/45/9/008}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/PericoliRidolfini2005.pdf:pdf;:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/PericoliRidolfini2005{\_}ValidationPAM{\_}FTU.pdf:pdf}, +issn = {0029-5515}, +journal = {Nuclear Fusion}, month = {sep}, -number = {9A}, -pages = {1365--1376}, -title = {{Review of lower hybrid wave heating and current drive}}, -url = {http://stacks.iop.org/0741-3335/28/i=9A/a=014?key=crossref.2f7fb0fd445d2b2e047c5503ff107988}, -volume = {28}, -year = {1986} +number = {9}, +pages = {1085}, +title = {{LHCD and coupling experiments with an ITER-like PAM launcher on the FTU tokamak}}, +url = {http://stacks.iop.org/0029-5515/45/i=9/a=008 http://stacks.iop.org/0029-5515/45/i=9/a=008?key=crossref.6b9bde59e11fd1ed8c9b8be441b530ec}, +volume = {45}, +year = {2005} +} +@article{Bernabei1975, +author = {Bernabei, S. and Heald, M A and Hooke, W M and Paoloni, F J}, +doi = {10.1103/PhysRevLett.34.866}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Bernabei1975{\_}Penetration of Slow Waves into a Dense Plasma Using a Phased Wave-Guide Array.pdf:pdf}, +journal = {Phys. Rev. Lett.}, +number = {14}, +pages = {866--870}, +publisher = {American Physical Society}, +title = {{Penetration of Slow Waves into a Dense Plasma Using a Phased Wave-Guide Array}}, +url = {http://link.aps.org/doi/10.1103/PhysRevLett.34.866}, +volume = {34}, +year = {1975} } -@techreport{Bers1981, -author = {Bers, A and Theilhaber, K S}, -file = {::}, -institution = {Plasma Fusion Center, MIT, Cambridge, MA}, -number = {PFC/JA-81-20}, -title = {{Linear Theory of Coupling EM Power from Waveguide Arrays to a Plasma in a Magnetic Field}}, -year = {1981} +@booklet{Tenenbaum2003, +author = {Tenenbaum, Peter}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Tenenbaum2003{\_}A Brief Introduction to RF Power Sources.pdf:pdf}, +title = {{A Brief Introduction to RF Power Sources Klystrons}}, +url = {http://www.desy.de/{~}njwalker/uspas/}, +year = {2003} } -@article{Vladimira, -author = {Vladimir, Krapchev and Krapchev, Vladimir B.}, -doi = {10.1103/PhysRevLett.42.497}, -issn = {0031-9007}, -journal = {Physical Review Letters}, -month = {feb}, -number = {8}, -pages = {497--500}, -title = {{Kinetic Thoery of the Pondermotive Effects in Plasma}}, -url = {https://link.aps.org/doi/10.1103/PhysRevLett.42.497}, -volume = {42}, -year = {1979} +@article{Porkolab1984a, +author = {Porkolab, Miklos}, +doi = {10.1109/TPS.1984.4316303}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Porkolab1984.pdf:pdf}, +issn = {0093-3813}, +journal = {IEEE Transactions on Plasma Science}, +number = {2}, +pages = {107--117}, +title = {{Survey of Lower Hybrid Experiments}}, +url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=4316303}, +volume = {12}, +year = {1984} } @article{Stevens1988, abstract = {Coupling structures for lower hybrid current drive experiments have, until now, been smaller than a free space wavelength and have had a correspondingly broad wavenumber spectrum. The paper reports the results of experiments on the PLT tokamak using a 16-waveguide grill (2.2 wavelengths) which produces a very narrow n ? = k ? c/? spectrum. Experimental results from the 16-waveguide grill are compared with results from three other PLT grills with less sharply defined n 1 spectra. The current drive figure of merit, {\#}{\#}IMG{\#}{\#} [http://ej.iop.org/images/0029-5515/28/2/004/nf{\_}28{\_}2{\_}004inline1.gif] , is approximately 40{\%} higher for the experiments with the 16-waveguide coupler than for previously reported experiments on PLT, in spite of the larger 'spectral gap'. The experimental results are consistent with the first-pass damping of a large fraction of the launched spectrum.}, author = {Stevens, J. E. and Bell, R. E. and Bernabei, S. and Cavallo, a. and Chu, T. K. and Colestock, P. L. and Hooke, W. and Hosea, J. and Jobes, F. and Luce, T. and Mazzucato, E. and Motley, R. and Pinsker, R. and Goeler, S Von and Wilson, J. R. and {Von Goeler}, S. and Wilson, J. R.}, doi = {10.1088/0029-5515/28/2/004}, -file = {::}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Stevens1988{\_}Lower hybrid experiments on PLT using grills with various n{\_}parallel spectral widths.pdf:pdf}, isbn = {0029-5515}, issn = {00295515}, journal = {Nuclear Fusion}, @@ -934,47 +952,133 @@ @article{Stevens1988 volume = {28}, year = {1988} } -@book{Pond2008, -author = {Pond, N H}, -isbn = {9780981692302}, -publisher = {Russ Cochran, Publisher}, -title = {{The tube guys}}, -url = {http://books.google.fr/books?id=fXEfAQAAIAAJ}, +@phdthesis{Karney1977b, +author = {Karney, Charles F. F.}, +file = {:U$\backslash$:/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Karney1978{\_}Stochastic ion heating by a lower hybrid wave.pdf:pdf}, +school = {MIT}, +title = {{Stochastic heating of Ions in a Tokamak by RF Power}}, +year = {1977} +} +@article{Hooke1982, +author = {Hooke, W and Others}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Hooke1983{\_}Lower Hybrid Heating and Current Drive on PLT.pdf:pdf}, +institution = {PPPL-1976}, +title = {{Lower Hybrid Heating and Current Drive on {\{}PLT{\}}}}, +year = {1983} +} +@book{Jackson1998, +author = {Jackson, John David}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Ebooks/Electromagnetisme/Jackson/Jacksons Classical Electrodynamics (Solutions).pdf:pdf}, +publisher = {Wiley, New York}, +title = {{Classical Electrodynamics}}, +year = {1998} +} +@book{Benford2007a, +author = {Benford, James and Swegle, John A and Schamiluglu, Edl}, +edition = {second edi}, +editor = {CRC}, +isbn = {9780750307062}, +pages = {552}, +title = {{High Power Microwaves}}, +year = {2007} +} +@article{Gormezano1984a, +abstract = {The experimental data of the HF coupling measurements of the 4 waveguide grill of the Wega Tokamak have been compared with a linear coupling theory using a step density model. In order to minimize specific boundary effects which are not taken into account in the theory, the authors made use of data obtained when only the central waveguides are fed. The plasma density and its gradient at the mouth of the grill are estimated from probe measurements made with plasma conditions similar to those of the experimental coupling data. The qualitative agreement is always very good and a quantitative agreement is obtained in a relatively high density regime. The validity of the step density model is supported by the density dependence of the phases of the reflected signals.}, +author = {Gormezano, C. and Moreau, D}, +doi = {10.1088/0741-3335/26/3/005}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Gormezano1984.pdf:pdf}, +isbn = {0741-3335}, +issn = {0741-3335}, +journal = {Plasma Physics and Controlled Fusion}, +month = {mar}, +number = {3}, +pages = {553}, +title = {{Lower hybrid wave coupling in the Wega Tokamak}}, +url = {http://stacks.iop.org/0741-3335/26/i=3/a=005 http://stacks.iop.org/0741-3335/26/i=3/a=005?key=crossref.b0988de8436642feb6abfe45dbf08c33}, +volume = {26}, +year = {1984} +} +@article{Koert2008a, +author = {Koert, P and MacGibbon, P and Vieira, R and Terry, D and Leccacorvi, R and Doody, J and Beck, W}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Koert2008{\_}Waveguide Splitter for LHCD.pdf:pdf}, +institution = {PSFC/JA-08-50}, +title = {{Waveguide Splitter for Lower Hybrid Current Drive}}, +url = {http://www.new.ans.org/pubs/journals/fst/a{\_}8885}, year = {2008} } -@article{Litaudon1992a, -abstract = {The TORE SUPRA lower hybrid current drive experiments (8 MW/3.7 GHz) use large phased waveguide arrays, four rows of 32 active waveguides and two passive waveguides for each of the two grills, to couple the waves to the plasma. These launchers are based on the 'multijunction' principle which allows them to be quite compact and is therefore attractive for the design of efficient multi-megawatt antennas in NET/ITER. Extensive coupling measurements have been performed in order to study the radiofrequency (RF) characteristics of the plasma loaded antennas. Measurements of the plasma scattering coefficients of the antennas show good agreement with those obtained from the linear coupling theory (SWAN code). Global reflection coefficients of a few per cent have been measured in a large range of edge plasma densities (0.3 x 10(18) m-3 less-than-or-equal-to n(eg) less-than-or-equal-to 1.4 x 10(18) m-3) or antenna positions (0.02-0.05 m from the plasma edge) and up to a maximum injected RF power density of 45 MW/m2. When the plasma is pushed against the inner wall of the chamber, the reflection coefficient is found to remain low up to distances of the order of 0. 10 m. The coupling measurements allow us to deduce the 'experimental' power spectra radiated by the antennas when all their modules are fed simultaneously with variable phases. Thus, the multijunction launcher is assessed as a viable antenna for high power transmission with good coupling characteristics and spectrum control.}, -author = {Litaudon, X and Berger-by, G and Bibet, Philippe and Bizarro, Jo{\~{a}}o P.S. and Capitain, J J and Carrasco, J and Goniche, M and Hoang, G T and Kupfer, K and Magne, R. and Moreau, D and Peysson, Yves and Rax, J.-M. M and Rey, G and Rigaud, D and Tonon, G and Bergerby, G and Bibet, Philippe and Bizarro, Jo{\~{a}}o P.S. and Capitain, J J and Carrasco, J and Goniche, M and Hoang, G T and Kupfer, K and Magne, R. and Moreau, D and Peysson, Yves and Rax, J.-M. M and Rey, G and Rigaud, D and Tonon, G}, -doi = {10.1088/0029-5515/32/11/I01}, -file = {::}, -isbn = {0029-5515}, -issn = {00295515}, +@article{Litaudon1990a, +author = {Litaudon, X and Moreau, D}, +doi = {10.1088/0029-5515/30/3/009}, +issn = {0029-5515}, journal = {Nuclear Fusion}, -keywords = {current drive,frequency,grill,jet}, -number = {11}, -pages = {1883--1897}, -title = {{Lower Hybrid Wave Coupling in Tore Supra through Multijunction Launchers}}, -url = {http://stacks.iop.org/0029-5515/32/i=11/a=I01}, -volume = {32}, -year = {1992} +month = {mar}, +number = {3}, +pages = {471--484}, +title = {{Coupling of slow waves near the lower hybrid frequency in JET}}, +url = {http://stacks.iop.org/0029-5515/30/i=3/a=009?key=crossref.1b8efc2102e93db4c4a8a0cdc2275c12}, +volume = {30}, +year = {1990} } -@inproceedings{Motley1985, -abstract = {A study of radiofrequency current ramp-up in the PLT tokamak is reported. The plasma current was first raised to 200–300 kA by the Ohmic heating transformer, and the current in the transformer primary circuit was then held constant to remove the OH drive. After the current fell below 200 kA, up to 300 kW of toroidally-directed RF power at 800 MHz was transmitted into the PLT plasma via a 6-element phased waveguide array. Current ramp-up rates between 0 and 120 kA/s for a 0.35 s time interval ((½–1/3) L/R time) where measured at densities between 2 and 4 × 1012 cm−3. It is estimated that about 20{\%} of the RF energy introduced into the vacuum vessel was converted into poloidal magnetic field energy, LI2/2, where L ≅ 3 $\mu$H is the total inductance of the plasma current loop. This conversion ratio should depend on a variety of factors, including the percentage of RF power absorbed by resonant electrons and the magnitude of the back current induced by the changing poloidal flux LI. The high ramp-up efficiencies are predicted theoretically in the regime in which the PLT ramp-up experiments operate, i.e., where the phase velocity of the waves is approximately equal in magnitude to the runaway velocity due to the back voltage. Comparison of the raw data with theory suggest that about ½ to ¾ of the incident RF power is absorbed by resonant high-velocity electrons.}, -annote = {Paper IAEA--CN--44/F--II--2 -Proc. Tenth International Conf., London, England, Sept. 12--19, 1984}, -author = {Motley, Robert W and Bell, Ronald E and Bernabei, Stefano and Cavallo, Alfred J and Chu, Tsu-Kai and Cohen, Samuel A and Denne, Boel G and Efthimion, Phillip C and Fisch, Nathaniel J. and Hinnov, Einar and Hooke, William M and Hosea, Joel C and Jobes, Forrest C and Karney, Charles F. F. and Mazzucato, Ernesto and Meservey, E. and Stevens, James E and Suckewer, Szymon and Taylor, Gary and Timberlake, John R and von Goeler, Schweickhard E and Wilson, J Randall}, -booktitle = {Plasma Physics and Controlled Nuclear Fusion Research 1984}, -pages = {473--478}, -publisher = {IAEA, Vienna}, -title = {{Lower Hybrid Current Ramp-up in the {\{}PLT{\}} Tokamak}}, -url = {https://inis.iaea.org/search/search.aspx?orig{\_}q=RN:16032336}, -volume = {1}, -year = {1985} +@article{Motley1980, +author = {Motley, R W and Hooke, W M}, +doi = {10.1088/0029-5515/20/2/013}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Motley1980{\_}ActivePassiveWaveguideArrayForWaveExcitationInPlasmas.pdf:pdf;:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Motley1980{\_}ActivePassiveWaveguideArraysForWaveExcitationInPlasmas.pdf:pdf}, +issn = {17414326}, +journal = {Nuclear Fusion}, +pages = {222--224}, +title = {{Active-passive waveguide array for wave excitation in plasmas}}, +volume = {222}, +year = {1979} +} +@phdthesis{Meneghini2012, +author = {Meneghini, Orso}, +school = {Massachusetts Institute of Technology}, +title = {{Full-Wave modeling of lower hybrid waves on Alcator C-Mod}}, +year = {2012} +} +@article{Mirizzi2003, +abstract = {This paper outlines the preliminary radiofrequency analysis of the prominent microwave components of the LHCD system for ITER. The general overview and inclusive analysis of the system is given in Ph. Bibet et al. (Overview of the ITER-Feat LH System; this Conference), a companion paper in this Conference. The results of the analysis and optimisation of the most relevant components is reported as computed by the 'high frequency structure simulator' (HFSS), a computer code developed by ANSOFT and based on the finite elements method. ?? 2003 Elsevier Science B.V. All rights reserved.}, +author = {Mirizzi, F. and Bibet, Philippe and Kuzikov, S.}, +doi = {10.1016/S0920-3796(03)00081-4}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Mirizzi2003.pdf:pdf}, +isbn = {0920-3796}, +issn = {09203796}, +journal = {Fusion Engineering and Design}, +keywords = {Components,Microwave,Radiofrequency}, +pages = {487--490}, +title = {{The main microwave components of the LHCD system for ITER}}, +volume = {66-68}, +year = {2003} +} +@book{Faria2008, +abstract = {The material presented in the book is built on a substrate of knowledge already provided by the basic sciences of mathematics and physics. Students are supposed to be acquainted with certain topics, such as linear algebra, differential equations, integral calculus, vector analysis and complex functions. If students still have difficulties with these topics, they may have to recap them in order to refresh their skills. This book is not a treatise on electricity and magnetism – its scope is far less ambitious. Its content can be delivered in a single-semester course, and is aimed to provide a scientifically founded and unified basis of fundamental knowledge on electromagnetic field phenomena that will help students follow up more advanced subjects covered in their courses. Topics are introduced in a systematic and friendly manner, proceeding from the simpler to more difficult ones, using a slow build-up process. In addition, a series of application examples and homework problems have been prepared to help students through the learning process. The fact that the book is partitioned into chapters does not imply that some of them can be skipped. Because the subject matter is deeply interrelated, students must try to adhere to the normal chapter sequence, otherwise they may be wasting their time or fail to get an integrated comprehensive view of the electromagnetic phenomena.}, +author = {Faria, J. A Brand{\~{a}}o}, +booktitle = {Electromagnetic Foundations of Electrical Engineering}, +doi = {10.1002/9780470697498}, +isbn = {9780470727096}, +pages = {1--399}, +title = {{Electromagnetic Foundations of Electrical Engineering}}, +year = {2008} +} +@article{Gormezano1986, +abstract = {The author reviews the interaction of lower hybrid waves and plasmas which is a very versatile method. The method has proven to be effective in a large range of applications which the author discusses: (1) bulk ion heating; (2) bulk electron heating; and (3) noninductive current drive.}, +author = {Gormezano, C.}, +doi = {10.1088/0741-3335/28/9A/014}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Gormezano1986{\_}Review{\_}LHCD.pdf:pdf}, +issn = {0741-3335}, +journal = {Plasma Physics and Controlled Fusion}, +month = {sep}, +number = {9A}, +pages = {1365--1376}, +title = {{Review of lower hybrid wave heating and current drive}}, +url = {http://stacks.iop.org/0741-3335/28/i=9A/a=014 http://stacks.iop.org/0741-3335/28/i=9A/a=014?key=crossref.2f7fb0fd445d2b2e047c5503ff107988}, +volume = {28}, +year = {1986} } @article{Bernabei1982, author = {Bernabei, S. and Daughney, C. and Efthimion, P. and Hooke, W. and Hosea, J. and Jobes, F. and Martin, A. and Mazzucato, E. and Meservey, E. and Motley, R. and Stevens, J. and Goeler, S Von and Wilson, R.}, doi = {10.1103/PhysRevLett.49.1255}, -file = {::}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Bernabei1982{\_}Lower-Hybrid Current Drive in the PLT Tokamak.pdf:pdf}, isbn = {0031-9007}, issn = {0031-9007}, journal = {Physical Review Letters}, @@ -987,59 +1091,45 @@ @article{Bernabei1982 volume = {49}, year = {1982} } -@article{TONON1977, -author = {TONON, G. and BLANC, P. and GORMEZANO, C. and HESS, W. and ICHTCHENKO, G. and NGUYEN, T. K. and DURVAUX, M. and MAGNE, R. and OHLENDORF, W. and PACHER, G. and PACHER, H. and WEGROWE, J. G.}, -doi = {10.1051/jphyscol:1977615}, -file = {::}, -issn = {0449-1947}, -journal = {Le Journal de Physique Colloques}, -month = {dec}, -number = {C6}, -pages = {C6--161--C6--169}, -title = {{LOWER HYBRID FREQUENCY HEATING IN TOROIDAL DEVICES WITH EMPHASIS ON WEGA RESULTS}}, -url = {http://www.edpsciences.org/10.1051/jphyscol:1977615 http://dx.doi.org/10.1051/jphyscol:1977615}, -volume = {38}, -year = {1977} -} -@article{Puri1974, -abstract = {Previously, it has been shown that in a cold, inhomogeneous, magnetized plasma half-space the lower-hybrid resonance is accessible to the transverse-magnetic (TM) plane waves incident on the vacuum-plasma interface at an approximately grazing incidence, provided that at the hybrid layer ? pe /? ce ? 0.4. In this paper, these results are extended to the slow-wave case when n z , the refractive index in the static magnetic field direction, exceeds unity. It is found that the plasma is indeed accessible to the slow waves if Golant's accessibility criterion n z {\textgreater} 1 + (? pe /? ce ) 2 is satisfied. The following recommendations can be made for coupling r.f. energy to the lower-hybrid resonance: (i) if ? pe /? ce ? 0.4, efficient coupling is possible by launching TEM-like waves on the plasma column, (ii) if ? pe /? ce ? 0.4 and if the transverse machine dimensions exceed the r.f. vacuum wavelength, it is possible to couple TM waves using passive slow-wave structures inside the machine walls, (iii) if ? pe /? ce ? 0.4, but for smaller machine dimensions, recourse must be taken to transverse-electric slow-wave coupling with current-carrying coils of appropriate periodicity . If, as was pointed out by Glagolev, propagation from the plasma edge to the hybrid layer is not materially affected by the inclusion of finite-temperature effects, by far the most elegant solution (with potential application to thermonuclear plasmas) for coupling r.f. energy from the second to the twentieth ion-cyclotron harmonic waves is by launching TEM-like waves in the coaxial waveguide formed by the plasma columnand the containing walls.}, -author = {Puri, S and Tutter, M}, -doi = {10.1088/0029-5515/14/1/014}, -file = {::}, +@article{Theilhaber1980, +abstract = {Launching of the fast wave in the lower hybrid frequency range is described. This wave is excited at the plasma edge by RF electric fields perpendicular to those required for the lower hybrid wave. In high-temperature plasmas, where the lower hybrid wave may not penetrate because of Landau damping or other effects near the edge, the fast wave might provide an alternative for heating and/or current generation in the central portion of the plasma. In addition, for high-density plasmas, this has the advantage that lower frequencies than those required for the lower hybrid excitation can be used. Thus waveguides of convenient dimensions for maximum power transmission and ease of fabrication can be employed. Coupling from a waveguide array into an inhomogeneous plasma is analysed. The model is infinite in y, the direction perpendicular to magnetic field and density gradient. Power reflection in the waveguides is found as a function of array design and density gradient at the edge. This reflection is fairly large ({\textgreater} 20{\%}). Propagation into the plasma is then considered, and the field structure and dispersion of the fast waves are found as functions of the distance of penetration. Unlike the lower hybrid waves, fast waves do not form resonance cones and energy is dispersed over a large volume.}, +author = {Theilhaber, K and Bers, A}, +doi = {10.1088/0029-5515/20/5/003}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Theilhaber1980{\_}Coupling to the Fast Wave at LH frequencies.pdf:pdf}, issn = {0029-5515}, journal = {Nuclear Fusion}, -month = {jan}, -number = {1}, -pages = {93--101}, -title = {{Slow-wave coupling to the lower-hybrid resonance}}, -url = {http://stacks.iop.org/0029-5515/14/i=1/a=014 http://stacks.iop.org/0029-5515/14/i=1/a=014?key=crossref.f33f933ccc2480cfbaabffe38ddbffa5}, -volume = {14}, -year = {1974} -} -@book{Harrington2001, -author = {Harrington, Roger F}, -isbn = {978-0-471-20806-8}, -pages = {496}, -publisher = {Wiley-IEEE Press, New York}, -title = {{Time-Harmonic Electromagnetic Fields}}, -year = {2001} +month = {may}, +number = {5}, +pages = {547--555}, +title = {{Coupling to the fast wave at lower hybrid frequencies}}, +url = {http://stacks.iop.org/0029-5515/20/i=5/a=003?key=crossref.d24dafab0e5f71c54c1f342307d0b320}, +volume = {20}, +year = {1980} } -@inproceedings{Hoang2009, -abstract = {A 20 MW/5 GHz lower hybrid current drive (LHCD) system was initially due to be commissioned and used for the second mission of ITER, i.e. the Q = 5 steady state target. Though not part of the currently planned procurement phase, it is now under consideration for an earlier delivery. In this paper, both physics and technology conceptual designs are reviewed. Furthermore, an appropriate work plan is also developed. This work plan for design, R{\&}D, procurement and installation of a 20 MW LHCD system on ITER follows the ITER Scientific and Technical Advisory Committee (STAC) T13-05 task instructions. It gives more details on the various scientific and technical implications of the system, without presuming on any work or procurement sharing amongst the possible ITER partnersb The LHCD system of ITER is not part of the initial cost sharing.. This document does not commit the Institutions or Domestic Agencies of the various authors in that respect. {\textcopyright} 2009 IAEA, Vienna.}, -author = {Hoang, G.T. and B{\'{e}}coulet, A. and Jacquinot, J. and Artaud, J.F. J.-F. and Bae, Y.S. and Beaumont, B. and Belo, J.H. and Berger-By, G. and Bizarro, J.P.S. Jo{\~{a}}o P.S. and Bonoli, P. and Cho, M.H. and Decker, Joan and Delpech, L{\'{e}}na and Ekedahl, Annika and Garcia, J. and Giruzzi, G. and Goniche, M. and Gormezano, C. and Guilhem, D. and Hillairet, J. and Imbeaux, F. and Kazarian, Fabienne and Kessel, C. and Kim, S.H. and Kwak, J.G. and Jeong, J.H. and Lister, J.B. and Litaudon, X. and Magne, R. and Milora, S. and Mirizzi, F. and Namkung, W. and Noterdaeme, J.M. and Park, S.I. and Parker, R. and Peysson, Yves and Rasmussen, D. and Sharma, P.K. and Schneider, M. and Synakowski, E. and Tanga, A. and Tuccillo, A. and Wan, Y.X.}, -booktitle = {Nuclear Fusion}, -doi = {10.1088/0029-5515/49/7/075001}, -file = {:home/hash/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Hoang et al. - 2009 - A lower hybrid current drive system for ITER(3).pdf:pdf;:home/hash/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Hoang et al. - 2009 - A lower hybrid current drive system for ITER(4).pdf:pdf}, -issn = {0029-5515}, -month = {jul}, -number = {7}, -title = {{A lower hybrid current drive system for ITER}}, -url = {http://stacks.iop.org/0029-5515/49/i=7/a=075001?key=crossref.b48bdcea42107a5a9be1a7fb1e48a7c8}, -volume = {49}, -year = {2008} +@article{Brambilla1983, +abstract = {The authors have investigated quasi-linear ion Landau damping of lower hybrid waves in an inhomogeneous plasma. To this end, they have simultaneously solved Maxwell equations and the ion kinetic equation, starting from the antenna and proceeding towards the centre of a plane-layered plasma. As a consequence of the development of a suprathermal tail in the ion distribution function, the efficiency of the absorption increases and the absorption region is found to shift to lower densities as the launched power increases. Absorption is always complete at the layer where the wave phase velocity equals about three times the local ion thermal velocity, usually somewhat before the linear turning point is reached.}, +author = {Brambilla, Marco and Chen, Yan-ping}, +journal = {Nuclear Fusion}, +number = {4}, +pages = {541}, +title = {{Quasi-linear ion heating by lower hybrid waves}}, +url = {http://stacks.iop.org/0029-5515/23/i=4/a=013}, +volume = {23}, +year = {1983} } -@book{Harrington2001, -author = {Harrington, Roger F}, -publisher = {Wiley-IEEE Press, New York}, -title = {{Time-Harmonic Electromagnetic Fields}} +@article{Litaudon1992a, +abstract = {The TORE SUPRA lower hybrid current drive experiments (8 MW/3.7 GHz) use large phased waveguide arrays, four rows of 32 active waveguides and two passive waveguides for each of the two grills, to couple the waves to the plasma. These launchers are based on the 'multijunction' principle which allows them to be quite compact and is therefore attractive for the design of efficient multi-megawatt antennas in NET/ITER. Extensive coupling measurements have been performed in order to study the radiofrequency (RF) characteristics of the plasma loaded antennas. Measurements of the plasma scattering coefficients of the antennas show good agreement with those obtained from the linear coupling theory (SWAN code). Global reflection coefficients of a few per cent have been measured in a large range of edge plasma densities (0.3 x 10(18) m-3 less-than-or-equal-to n(eg) less-than-or-equal-to 1.4 x 10(18) m-3) or antenna positions (0.02-0.05 m from the plasma edge) and up to a maximum injected RF power density of 45 MW/m2. When the plasma is pushed against the inner wall of the chamber, the reflection coefficient is found to remain low up to distances of the order of 0. 10 m. The coupling measurements allow us to deduce the 'experimental' power spectra radiated by the antennas when all their modules are fed simultaneously with variable phases. Thus, the multijunction launcher is assessed as a viable antenna for high power transmission with good coupling characteristics and spectrum control.}, +author = {Litaudon, X and Berger-by, G and Bibet, Philippe and Bizarro, Jo{\~{a}}o P.S. and Capitain, J J and Carrasco, J and Goniche, M and Hoang, G T and Kupfer, K and Magne, R. and Moreau, D and Peysson, Yves and Rax, J.-M. M and Rey, G and Rigaud, D and Tonon, G and Bergerby, G and Bibet, Philippe and Bizarro, Jo{\~{a}}o P.S. and Capitain, J J and Carrasco, J and Goniche, M and Hoang, G T and Kupfer, K and Magne, R. and Moreau, D and Peysson, Yves and Rax, J.-M. M and Rey, G and Rigaud, D and Tonon, G}, +doi = {10.1088/0029-5515/32/11/I01}, +file = {:C$\backslash$:/Users/JH218595/Documents/Documents/Documents{\_}Bibliographie{\_}Ebooks{\_}Cours/Articles/Litaudon1992.pdf:pdf}, +isbn = {0029-5515}, +issn = {00295515}, +journal = {Nuclear Fusion}, +keywords = {current drive,frequency,grill,jet}, +number = {11}, +pages = {1883--1897}, +title = {{Lower Hybrid Wave Coupling in Tore Supra through Multijunction Launchers}}, +url = {http://stacks.iop.org/0029-5515/32/i=11/a=I01}, +volume = {32}, +year = {1992} } diff --git a/RF_Fundamentals.tex b/RF_Fundamentals.tex index d9d5b37..eddb494 100644 --- a/RF_Fundamentals.tex +++ b/RF_Fundamentals.tex @@ -1,37 +1,136 @@ \chapter{Radio-Frequency Fundamentals} - +% ########################################################################### +% ########################################################################### +% ########################################################################### \section{Introduction} In this book we will the International System (SI) of units, which differs from most of the plasma physics and waves into plasma reference books in which CGS units are used instead. In this system of units, the unit of length is the meter, the unit of time is the second and the unit of mass is the kilogram. This choice is motivated by the fact that most engineering tools such as electromagnetic solvers also use SI units by default. Moreover, these units are also the one used in practice when performing measurements. +% ########################################################################### +% ########################################################################### +% ########################################################################### \section{Basic Equations} +% ########################################################################### +% ########################################################################### \subsection{Maxwell Equations} We shall start the Maxwell equations is their most general form, before recasting them to fit our needs. The usual electromagnetic field quantities are expressed in terms of six quantities that are: \begin{itemize} \item $\mathcal{E}$: the electric field intensity (in $V/m$) \item $\mathcal{H}$: the magnetic field intensity (in $A/m$) - \item $\mathcal{D}$: the electric flux density (in $C/m^2$) - \item $\mathcal{B}$: the magnetic flux density (in $Wb/m^2$) + \item $\mathcal{D}$: the electric flux density (in $A\cdot s/m^2=C/m^2$) + \item $\mathcal{B}$: the magnetic flux density (in $V\cdot s/m^2=Wb/m^2$, also known as Tesla $T$) \item $\mathcal{J}$: the electric current density (in $A/m^2$) - \item $\mathcal{Q}$: the electric charge density (in $C/m^3$) + \item $q_v$: the electric charge density (in $C/m^3$) \end{itemize} where all quantities are function of space and time, e.g. $\mathcal{E}=\mathcal{E}(\mathbf{r},t)$. -Since James Clerk Maxwell discovered the full set of mathematical laws describing electromagnetic fields, many mathematicians, physicists and engineers have proposed different frameworks for representing fields and waves equations\parencite{Lindell2004, Warnick2014}. -For day-to-day work in electromagnetic engineering, Heaviside's vector representation is commonly used. Within this frame, Maxwell equations can be stated as a set of local differential equations: +Since James Clerk Maxwell discovered the full set of mathematical laws describing electromagnetic fields, many mathematicians, physicists and engineers have proposed different frameworks for representing fields and waves equations\parencite{Lindell2004, Warnick2014}. For day-to-day work in electromagnetic engineering, Heaviside's vector representation is commonly used. Within this frame, Maxwell equations can be stated as a set of local differential equations: \begin{subequations} \begin{align} \boldsymbol{\nabla} \times \boldsymbol{\mathcal{E}} &= -\frac{\partial \boldsymbol{\mathcal{B}}}{\partial t} \label{eq:Maxwell-Faraday}\\ \boldsymbol{\nabla} \times \boldsymbol{\mathcal{H}} &= \frac{\partial \boldsymbol{\mathcal{D}}}{\partial t} + \boldsymbol{\mathcal{J}} \label{eq:Maxwell-Ampere} \\ - \boldsymbol{\nabla} \cdot \boldsymbol{\mathcal{D}} &= \mathcal{Q} \label{eq:Maxwell-Gauss} \\ + \boldsymbol{\nabla} \cdot \boldsymbol{\mathcal{D}} &= q_v \label{eq:Maxwell-Gauss} \\ \boldsymbol{\nabla} \cdot \boldsymbol{\mathcal{B}} &= 0 \label{eq:Maxwell-Gauss-Magnetism} \end{align} + \label{eq:MaxwellEquations} \end{subequations} -The Maxwell-Faraday's law \ref{eq:Maxwell-Faraday} relates the magnetic flux to the electric field, by describing how a time-varying field induces an electric field. -The Maxwell-Ampere's law \ref{eq:Maxwell-Ampere} relates the current to the the magnetic field. It states that magnetic field can be generated by an electric current or by changing the electric field. The Maxwell-Gauss law \ref{eq:Maxwell-Gauss} describes the relationship between an electric flux density and the electric charges that cause it. The Maxwell-Gauss law for magnetism states that no magnetic charge exists as for electric charges. +The Maxwell-Faraday's law \ref{eq:Maxwell-Faraday} relates the magnetic flux to the electric field, by describing how a changing magnetic flux induces an electric field. +The Maxwell-Ampere's law \ref{eq:Maxwell-Ampere} relates the current to the magnetic field. It states that magnetic field can be generated by a changing electric flux density and by electric current. +The Maxwell-Gauss law \ref{eq:Maxwell-Gauss} describes the relationship between an electric flux density and the electric charges that cause it. +The Maxwell-Gauss law for magnetism states that no magnetic charge exists as for electric charges. + +The corresponding equations in integral form are: +\begin{subequations} + \begin{align} + \oint \boldsymbol{\mathcal{E}} \cdot \diff \mathbf{l} + =& + - \frac{\diff }{\diff t} \iint \boldsymbol{\mathcal{B}} \cdot \diff \mathbf{S} + \\ + \oint \boldsymbol{\mathcal{B}} \cdot \diff \mathbf{l} + =& + \frac{\diff }{\diff t} \iint \boldsymbol{\mathcal{D}} \cdot \diff \mathbf{S} + + \iint \boldsymbol{\mathcal{J}} \cdot \diff \mathbf{S} + \\ + \oiint \boldsymbol{\mathcal{B}} \cdot \diff \mathbf{S} + =& \; 0 + \\ + \oiint \boldsymbol{\mathcal{D}} \cdot \diff \mathbf{S} + =& + \iiint q_v \diff v + \end{align} + \label{eq:MaxwellEquationsIntegral} +\end{subequations} + +On can define the following \emph{circuit} quantities associated with each field quantities\parencite{Harrington2001}: +\begin{itemize} + \item $v$, the \emph{voltage} in $V$ + \item $i$, the \emph{current} in $A$ + \item $q$, the \emph{electric charge} in $C$ + \item $\psi$, the \emph{magnetic flux} in $Wb$ + \item $\psi_e$, the \emph{electric flux} in $C$ + \item $u$, the \emph{magnetomotive force} in $A$ +\end{itemize} +defined by: +\begin{subequations} + \begin{align} + v =& \int \boldsymbol{\mathcal{E}} \cdot \diff \mathbf{l} \\ + i =& \iint \boldsymbol{\mathcal{J}} \cdot \diff \mathbf{S} \\ + q =& \iiint q_v \diff v \\ + \psi =& \iint \boldsymbol{\mathcal{B}} \cdot \diff \mathbf{S} \\ + \psi_e=& \iint \boldsymbol{\mathcal{D}} \cdot \diff \mathbf{S} \\ + u =& \int \boldsymbol{\mathcal{H}} \cdot \diff \mathbf{l} + \end{align} + \label{eq:CircuitQuantities} +\end{subequations} -In order to be able to apply the previous equations, we need to specify the relationships existing between electric, magnetic flux densities ($\mathcal{D}$,$\mathcal{B}$) and electric current density ($\mathcal{J}$) with electric and magnetic intensities ($\mathcal{E},\mathcal{H}$). These relations depend on the medium properties in which the field exists and are called \emph{constitutive relationships}. +In order to be able to solve the equations(\ref{eq:MaxwellEquations}), one needs to specify the relationships existing between electric, magnetic flux densities ($\mathcal{D}$,$\mathcal{B}$) and electric current density $\mathcal{J}$ with electric and magnetic intensities ($\mathcal{E},\mathcal{H}$)\footnote{ + I've chosen here is to treat both $\mathbf{E}$ and $\mathbf{H}$ as similar kinds of fields, different from $\mathbf{D}$ and $\mathbf{B}$, as for example it is done in \parencite{Pozar1998} or \parencite{Harrington2001}. This choice has symbolic advantages when dealing with RF networks with SI units, where the electric and magnetic field intensity can be associated to voltages and currents as they are measured (tbc). + + However, many (most?) authors have chosen instead to treat $\mathbf{E}$ and $\mathbf{B}$ as the \emph{fundamental} quantities and $\mathbf{D}$ and $\mathbf{H}$ as \textit{auxiliary} (or convenience) quantities denoting the average fields over macroscopic small regions\parencite{Lindell1995, Griffiths2005, Jackson1999}. Indeed, both $\mathbf{D}$ and $\mathbf{H}$ allow to write Gauss and Ampere laws in terms of the \emph{free} charges and currents alone, such "incorporating" \emph{bounds} charge and current contributions. Thus, in these macroscopic Maxwell equations, only the external charges and currents brought into the system from outside are considered, without taking care of the average charge and current distributions in the medium, which can be a convenient mathematical tool. This choice has sense, this one cannot turn off bounds contributions as he can for free contributions \parencite[sec.6.3]{Griffiths2005}. In such as case, one would have read instead $\mathbf{H}=\boldsymbol{\mu}^{-1} $. Finally, the magnetic flux density $\mathbf{B}$ can be derived as the character of an electric field in a co-moving frame\parencite{Schwinger1998}. + + The latter choice can also take origin from the interpretation that the electric and magnetic fields as a force acting on a test charge via the Lorentz force $q(\mathbf{E}+\mathbf{v}\times\mathbf{B})$, leading naturally to $(\mathbf{E}$,$\mathbf{B})$. However, if one uses an energy picture using differential forms, interpreting the electromagnetic field as the change of energy experienced by a test charge as it moves through the field, the couples $(\mathbf{E}$,$\mathbf{H})$ and $(\mathbf{D}$,$\mathbf{B})$ are represented by different mathematical objects\parencite{Warnick2014}. + } +. These relations depend on the medium properties in which the field exists and are called \emph{constitutive relations}. Explicit forms of these relationships have been found from experimentation or deduced from atomic considerations \parencite[sec.5]{Schwinger1998} and are discussed in the next section. + +% ########################################################################### +% ########################################################################### +\subsection{Constitutive Relations} +Fluxes densities ($\mathcal{D}$,$\mathcal{B}$) differ from field intensities ($\mathcal{E},\mathcal{H}$) inside the material with regards to relative magnitude and direction. Flux densities can be interpreted as a response of the medium to an applied excitation\footnote{If we recall the Gauss law $ Q = \oint \boldsymbol{\mathcal{D}} \cdot \diff S$, the flux $\boldsymbol{\mathcal{D}}$ depends on the charge inside the closed surface and doesn't depend on the material itself, but the field intensity does. } +. Such, the constitutive relationships can be written generally as: +\begin{subequations} + \begin{align} + \boldsymbol{\mathcal{D}} =& \boldsymbol{\mathcal{D}}(\boldsymbol{\mathcal{E}},\boldsymbol{\mathcal{H}}) \\ + \boldsymbol{\mathcal{B}} =& \boldsymbol{\mathcal{B}}(\boldsymbol{\mathcal{E}},\boldsymbol{\mathcal{H}}) \\ + \boldsymbol{\mathcal{J}} =& \boldsymbol{\mathcal{J}}(\boldsymbol{\mathcal{E}},\boldsymbol{\mathcal{H}}) + \end{align} +\end{subequations} +All of these relations hold only if the time rate of change of the electromagnetic field is small enough. Otherwise, one needs to extend the definition of linearity using linear differential relations\parencite{Harrington2001, Jackson1998}: +\begin{subequations} + \begin{align} + \boldsymbol{\mathcal{D}} &= \varepsilon \boldsymbol{\mathcal{E}} + \varepsilon_1 \frac{\partial \boldsymbol{\mathcal{E}}}{\partial t} + \varepsilon_2 \frac{\partial^2 \boldsymbol{\mathcal{E}}}{\partial t^2} + \ldots \\ + \boldsymbol{\mathcal{B}} &= \varepsilon \boldsymbol{\mathcal{H}} + \varepsilon_1 \frac{\partial \boldsymbol{\mathcal{H}}}{\partial t} + \varepsilon_2 \frac{\partial^2 \boldsymbol{\mathcal{H}}}{\partial t^2} + \ldots + \end{align} +\end{subequations} +Such situation arises typically when high intensity RF fields are used, which leads to non-linear phenomenons such \emph{ponderomotive effect}\parencite{Krapchev1979}. + + +%% ########################################################################### +%\subsubsection{General linear medium} +%\begin{subequations} +% \begin{align} +% \mathbf{D} =& \boldsymbol{\varepsilon}\mathbf{E} + \mathbf{P} \\ +% \mathbf{B} =& \boldsymbol{\mu}\left( \mathbf{H} + \mathbf{M} \right) +% \end{align} +%\end{subequations} +%where: +%\begin{itemize} +% \item $\mathbf{P}$ is the electric polarization of the medium, caused by displacement of bounds charges, measured in $C/m^2$. +% \item $\mathbf{M}$ is the magnetisation of the medium (or magnetic polarization), which corresponds to the distribution of magnetic moments per unit volume, measured in $A/m$\footnote{Note that $\mathbf{M}$ is included in the parenthesis, while $\mathbf{P}$ is not, for a matter of historical definition. This of course affects the units of these quantities. }. +%\end{itemize} + +% ########################################################################### +\subsubsection{Vacuum} In vacuum or in any other medium having similar characteristics than vacuum (such as air), the constitutive relationships have their most simpler form: \begin{subequations} \begin{align} @@ -42,6 +141,8 @@ \subsection{Maxwell Equations} \end{subequations} where $\varepsilon_0$ is the vacuum \emph{permittivity} and $\mu_0$ the vacuum \emph{permeability}. +% ########################################################################### +\subsubsection{Isotropic linear mediums} In a standard isotropic linear mediums, the constitutive relationships becomes linear relationships: \begin{subequations} \begin{align} @@ -56,7 +157,9 @@ \subsection{Maxwell Equations} The medium permittivity $\varepsilon$ can never be less than the vacuum permittivity $\varepsilon_0$. The \emph{relative permittivity} is defined such as $\varepsilon_r=\varepsilon/\varepsilon_0$. The permittivity of a conductor is hard to measure but appears to be unity\parencite{Harrington2001}. A similar definition holds for the \emph{relative permeability} $\mu_r=\mu/\mu_0$. For almost all materials except \emph{ferromagnetic} materials, one has $\mu=\mu_0$. -In anisotropic linear mediums, the constitutive relationships becomes tensor-relationships: +% ########################################################################### +\subsubsection{Anisotropic mediums} +If the response of the medium is different depending on the direction of the oscillating field, then the medium is called \emph{anisotropic}. In this case, his response is expressed by tensor relationships. In anisotropic linear mediums, the constitutive relationships becomes tensor-relationships: \begin{subequations} \begin{align} \boldsymbol{\mathcal{D}} &= \boldsymbol{\varepsilon} \cdot \boldsymbol{\mathcal{E}} \\ @@ -66,39 +169,112 @@ \subsection{Maxwell Equations} \end{subequations} where $\boldsymbol{\varepsilon}$, $\boldsymbol{\mu}$ and $\boldsymbol{\sigma}$ are the dielectric tensor, the permeability tensor and the conductivity tensor respectively, which can be interpreted as 3x3 matrices\parencite{Swanson2003}. -In general, the electromagnetic response of a medium is non-local with respect to both space and time\parencite{Mackay2010, Brambilla1998}. The medium response at the location $\mathbf{r}$ and time $t$ does not only depends on the field at location $\mathbf{r}$ and time $t$, but of the field in its vicinity $\mathbf{r}'$ and by all previous instant $t'$. Spacial non-locality can be significant when the wavelength is comparable to some characteristic length–scale in the medium. In plasma, the thermal agitation of the species induces add an additional erratic motion to the particles trajectory. Thus the particles are influenced by the field in the domain explored by their motion. In this situation, the constitutive relations of a linear medium should be stated as: +% ########################################################################### +\subsubsection{Nonlocal medium} +If a medium exhibits a time or space dependence to an electromagnetic excitation, it is said to be \emph{nonlocal} or \emph{dispersive}, with respect to time and space respectively. In a time and is called \emph{time dispersive} medium, explicit time dependence arises as a time delay between the imposition of the electric field and the resulting polarization of the medium. This delay is due to the inertia of charged particles to respond to the time-varying field\parencite{Mackay2010, Brambilla1998}. In a \emph{space-dispersive} medium, the response at the location $\mathbf{r}$ and time $t$ can not only depends on the field at location $\mathbf{r}$ and time $t$, but of the field in its vicinity $\mathbf{r}'$ and by all previous instant $t'$. Spatial non-locality can be significant when the wavelength is comparable to some characteristic length–scale in the medium. In plasma, the thermal agitation of the species induces add an additional erratic motion to the particles trajectory. Thus, the particles are influenced by the field in the domain explored by their motion\parencite{Brambilla1998}. This space dispersion can be omitted in the limit at which the temperature effects can be neglected. + +Thus, the constitutive relations of a general non-local anisotropic linear medium should be stated as: \begin{subequations} - \begin{align} - \boldsymbol{\mathcal{D}} = \int_{t'}\int_{\mathbf{r}'} &\left[ - \boldsymbol{\varepsilon}(\mathbf{r}', t') \cdot \boldsymbol{\mathcal{E}}(\mathbf{r},\mathbf{r}', t,t') + \right.\\ - & \left. \boldsymbol{\nu}(\mathbf{r}', t') \cdot \boldsymbol{\mathcal{H}}(\mathbf{r},\mathbf{r}', t,t') \right]d\mathbf{r}' dt' \nonumber \\ - \boldsymbol{\mathcal{B}}= \int_{t'}\int_{\mathbf{r}'} &\left[ - \boldsymbol{\xi}(\mathbf{r}', t') \cdot \boldsymbol{\mathcal{E}}(\mathbf{r},\mathbf{r}', t,t') + \right.\\ - & \left. \boldsymbol{\mu}(\mathbf{r}', t') \cdot \boldsymbol{\mathcal{H}}(\mathbf{r},\mathbf{r}', t,t') \right] d\mathbf{r}' dt' \nonumber - \end{align} + \begin{align} + \boldsymbol{\mathcal{D}}(\mathbf{r}, t) + = & + \int_{t'=-\infty}^t \diff t' + \int \diff \mathbf{r}' \; + \boldsymbol{\varepsilon}(\mathbf{r},\mathbf{r}', t,t') \cdot \boldsymbol{\mathcal{E}})(\mathbf{r}', t') + \\ + \boldsymbol{\mathcal{B}}(\mathbf{r}, t) + =& + \int_{t'=-\infty}^t \diff t' + \int \diff \mathbf{r}' \; + \boldsymbol{\mu}(\mathbf{r},\mathbf{r}', t,t') \cdot \boldsymbol{\mathcal{H}}(\mathbf{r}', t') + \\ + \boldsymbol{\mathcal{J}}(\mathbf{r}, t) + =& + \int_{t'=-\infty}^t \diff t' + \int \diff \mathbf{r}' \; + \boldsymbol{\sigma}(\mathbf{r},\mathbf{r}', t,t') \cdot \boldsymbol{\mathcal{E}}(\mathbf{r}', t') + \end{align} \end{subequations} +The restriction of time integration to times $t'