
\noindent Using the index of refraction~(\ref{indices_refraction}) the 
dispersion relation~(\ref{dispersion}) reads 
$$ \omega(k)\ =\ \frac{c\,k}{n(k)}\,.$$
Now, for the group velocity
\begin{eqnarray} \nonumber
  v_g = {d\omega\over dk} = \frac{c\,n}{n^2} - \frac{c\,k}{n^2}\,
  \frac{dn}{d\omega}\,\frac{d\omega}{dk}
  &\Rightarrow& n^2\,v_g + c\,k\,n'\,v_g = c\,n ~~{\rm with}~~
  n'=\frac{dn}{d\omega}\,. \\ \nonumber
  v_g &=& \frac{c\,n}{n^2+c\,k,n'}\,.
\end{eqnarray} 
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