+ \item \begin{itemize}
+ \item Equation of motion: $m\frac{d^2x}{dt^2}=-eE$
+ \item Ansatz: $x(t)=x_0 e^{-i\omega t}$
+ \item Solution of the equation of motion: $-m\omega^2x=-eE$
+ \item Dipole moment: $p=-ex=\frac{-e^2E}{m\omega^2}$
+ \item Polarization: $P=np=\frac{-ne^2E}{m\omega^2}$
+ \item Dielectric function:
+ $\epsilon(\omega)=1+\frac{P}{\epsilon_0E}
+ =1+\frac{-ne^2E}{m\omega^2\epsilon_0E}
+ =1-\frac{ne^2}{\epsilon_0m\omega^2}$
+ \end{itemize}
+ \item \begin{itemize}
+ \item Using $\omega_p^2=\frac{ne^2}{\epsilon_0m}$\\
+ $\Rightarrow \epsilon(\omega)=1-\frac{\omega_p^2}{\omega^2}$
+ \item Sketch of dielectric function:\\
+ (page 2)
+ \item Influence on electromagnetic waves:\\
+ $\frac{\omega}{\omega_p}>1\Leftrightarrow \omega>\omega_p$:
+ $\Rightarrow \epsilon=n^2>0$
+ $\Rightarrow$ transparent region\\
+ $\frac{\omega}{\omega_p}<1\Leftrightarrow \omega<\omega_p$:
+ $\Rightarrow \epsilon=n^2<0$
+ $\Rightarrow$ reflective region\\
+ \end{itemize}
+\input{dielectric_pslatex}