follow from Eq. (\[ImDrude\]) with an additional polarization term ${\cal P}$ in $\varepsilon
^{\prime }$:

$$\varepsilon ^{\prime }\left( \omega \right) ={\cal P}-\frac{\omega
_{p}^{2}}{\omega ^{2}+\omega _{\tau }^{2}},\ \ \varepsilon ^{\prime
\prime }\left( \omega \right) =\frac{\omega _{p}^{2}\omega _{\tau
}}{\omega \left( \omega ^{2}+\omega _{\tau }^{2}\right) }.
\label{DrudeRI}$$

The polarization term appears here due to the following reason. The total dielectric function $\varepsilon
=\varepsilon _{\left( c\right) }+\varepsilon _{\left( i\right) }$ includes contributions due to conduction electrons $\varepsilon
_{\left( c\right) }$ and the interband transitions $\varepsilon
_{\left( i\right) }$. The polarization term consists of the atomic polarizability and polarization due to the interband transitions $
\varepsilon _{\left( i\right) }^{\prime }$

$${\cal P}=1+\frac{N_{a}\alpha }{\varepsilon _{0}}+\varepsilon _{\left(
i\right) }^{\prime }\left( \omega \right) ,  \label{polariz}$$

where $\alpha $ is the atomic polarizability and $N_{a}$ the concentration of atoms. If the transition from intra- to interband absorption is sharp, the polarization can be considered as constant, because the interband transitions have a threshold behavior with an onset frequency $\omega _{0}$ and the Kramers-Kronig relation allows one to express $\varepsilon _{\left(
i\right) }^{\prime }$ as

$$\varepsilon _{\left( i\right) }^{\prime }\left( \omega \right) =\frac{2}{\pi
}\int\limits_{\omega _{0}}^{\infty }dx\frac{x\varepsilon _{\left( i\right)
}^{\prime \prime }\left( x\right) }{x^{2}-\omega ^{2}}.  \label{KKi}$$

For $\omega \ll \omega _{0}$ 