%info@http:://golem.fjfi.cvut.cz/wiki/root/GW4reports

\def\GWslide{\slide{Electron density $n_e$ interferometer measurement introduction (extreme simplification $B=0$)}{
\bc
Electromagnetic transverse wave:\\ $E_x(z,t)=E_0\cos(\omega t+kz)$
\twocolumns
{0.5}{
\underline{Maxwell Equations@Vacuum:}
$\begin{array}{ccc} \nabla \cdot  \vec E  & = & 0 \\ \nabla \cdot  \vec B  & = & 0 \\ \nabla \times \vec E  & = & - \partial_t \vec B \\ \nabla \times \vec B 	& = & \mu_0 \epsilon_0 \partial_t \vec E \end{array}$\\
$\Longrightarrow$\\
%Dispersion relation: $\omega^2=k^2c^2$: 
The refractive index: $N=1$\\
}
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\underline{Maxwell Equations@Plasma:}
$\begin{array}{ccc} \nabla \cdot  \vec E  & = & \rho/\epsilon_0 \\ \nabla \cdot  \vec B  & = & 0 \\ \nabla \times \vec E  & = & - \partial_t \vec B \\ \nabla \times \vec B 	& = & \mu_0 \vec J + \mu_0 \epsilon_0 \partial_t \vec E \end{array}$\\
$\Longrightarrow$
The refractive index: $N=\sqrt{1-\frac{{\omega _p}^2}{\omega ^2}}\approx 1-\frac{ne^2}{2\varepsilon_0m\omega^2}$
%Dispersion relation: $\omega^2=k^2c^2+w_{pe}^2$: 
}
The phase shift:\\ $\Delta\varphi=-\frac{e^2}{2c\varepsilon_0 m\omega}\int_0^L n(l) \, dl$
\ec
\credit{underthehood.blogwyrm:PlasmaWaves, Bartels:IPPlectures, MatenaMT}{}
}}