= Solution
At normal incidence from vacuum the <reflectance>[reflectivity] is
$$
R=\left|\frac{\sqrt{\epsilon_r}-1}{\sqrt{\epsilon_r}+1}\right|^2,
$$
with the passive branch of the <complex refractive index>. Below the optical resonance the real <permittivity> rises towards its pole and <reflectance> increases. Between the TO and LO frequencies, the lossless <permittivity> is negative; the wave is evanescent and the <reflectance>[reflectivity] is unity in that idealization. Finite <damping> rounds and reduces this <reststrahlen band>. Above the LO edge, positive <permittivity> returns and <reflectance>[reflectivity] can become very small when the <refractive index> is near one, before rising towards its electronic-background value.
Reading the surviving source graph to its available precision gives
$$
\boxed{\hbar\omega_{\rm TO}\simeq45\,\mathrm{meV},\quad
\hbar\omega_{\rm LO}\simeq50\,\mathrm{meV},\quad
\omega_{\rm TO}\simeq6.84\times10^{13}\,\mathrm{s^{-1}},\quad
\omega_{\rm LO}\simeq7.60\times10^{13}\,\mathrm{s^{-1}}.}
$$
These are band-edge estimates, not a high-precision <damping> fit. The minimum immediately above the band is not exactly the LO zero: for zero <damping> $\epsilon_r=1$ occurs at $\omega^2=(\epsilon_\infty\omega_{\rm LO}^2-\omega_{\rm TO}^2)/(\epsilon_\infty-1)>\omega_{\rm LO}^2$.
The high-<energy> plotted tail near $90\,\mathrm{meV}$ has $R\simeq0.22$. Ignoring residual <phonon> dispersion gives $n\simeq(1+\sqrt R)/(1-\sqrt R)=2.77$ and $\epsilon\simeq7.65$. Correcting with the finite-<energy> <oscillator> ratio gives
$$
\boxed{\epsilon_\infty\simeq7.65\frac{90^2-45^2}{90^2-50^2}\simeq8.3.}
$$
Thus a background of order $8$–$9$ is supported; equating the still-dispersive tail exactly to the asymptotic background would slightly underestimate it. The original model curves below use these approximate energies, without copying or fitting individual source data points.
\Image[/past-exam-of-the-condensed-matter-physics-major-option-course-of-the-university-of-oxford/2018-phonon-polariton.png]
{title=Infrared reflection and the distinct transverse polariton and longitudinal optical mode}
{description=Original oscillator-model curves with illustrative damping. Polariton branches are calculated independently of the measured points; the flat LO line is not the upper transverse polariton branch.}
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