$${\mathrm{Re}(\tilde n^2 )= n^2- \kappa^2= 1+ \frac{Ne^2}{\varepsilon_0 m_e} \frac{\omega_0^2 - \omega^2}{(\omega_0^2 - \omega^2 )^2+(\omega \gamma)^2}\\
\mathrm{Im}(\tilde n^2 ) = 2n\kappa= \frac{Ne^2}{\varepsilon_0 m_e} \frac{\omega \gamma}{(\omega_0^2 - \omega^2 )^2+(\omega \gamma)^2 }}$$