= Solution
Set $D=(\omega_0^2-\omega^2)^2+\gamma^2\omega^2$. The <real part> and <imaginary part> of the <complex permittivity> are
$$
\boxed{\epsilon_r'=1+\frac{\omega_p^2(\omega_0^2-\omega^2)}D,\qquad
\epsilon_r''=\frac{\omega_p^2\gamma\omega}D.}
$$
For positive frequency the loss part is nonnegative. The real part starts at $1+\omega_p^2/\omega_0^2$, rises below resonance, falls steeply through the resonance region, and approaches one from below at high frequency. The imaginary part is a positive absorption peak near $\omega_0$, tending to zero on both sides.
For a nonmagnetic medium write its <complex refractive index> as $N=n_r+i\kappa$, with $N^2=\epsilon_r$ and <optical extinction coefficient> $\kappa$. In a transparent region with $\kappa\ll n_r$ and $\epsilon_r'>0$,
$$
\boxed{n_r\simeq\sqrt{\epsilon_r'},\qquad
\kappa\simeq\frac{\epsilon_r''}{2\sqrt{\epsilon_r'}},\qquad
\alpha_{\rm abs}=\frac{2\omega\kappa}{c}.}
$$
The real part controls <phase velocity> and <optical dispersion>; the loss part controls <wave absorption> and the cycle-averaged dissipated power density $\omega\epsilon_0\epsilon_r''|E_0|^2/2$. Weak damping gives a narrow line but does not make <wave absorption> weak at its centre. The transparent-medium approximation applies away from the line.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2022-q4-optical-response.png]
{title=Single-resonance, multiple-resonance and free-electron optical responses}
{description=Original dimensionless model plots. The first row shows one bound-electron resonance, the second shows three, and the third shows the Drude permittivity and its normal-incidence reflectance. Finite damping is retained; these are illustrative calculations, not material measurements.}
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