= Solution
In the <Lorentz oscillator model>, each participating bound <Electron> is displaced by an applied <electric field> against a restoring force. The model replaces complicated <interband transitions> by a driven <damped harmonic oscillator>. If $u$ is the <Electron> displacement along the field and the <Electron> charge is $-e$, its equation of motion is
$$
m\ddot u+m\gamma\dot u+m\omega_0^2u=-eE(t).
$$
Here $\omega_0$ describes binding and $\gamma>0$ describes loss of coherent motion. For $E(t)=\operatorname{Re}(E_0e^{-i\omega t})$, the <complex amplitude> is
$$
u_0=-\frac{eE_0}{m(\omega_0^2-\omega^2-i\gamma\omega)}.
$$
The induced <electric dipole moment> is $-eu$, so the <electric polarization> amplitude for <number density> $n$ is
$$
P_0=\frac{ne^2}{m}\frac{E_0}{\omega_0^2-\omega^2-i\gamma\omega}.
$$
Using $D=\epsilon_0E+P=\epsilon_0\epsilon_rE$ gives
$$
\boxed{\epsilon_r(\omega)=1+\frac{\omega_p^2}{\omega_0^2-\omega^2-i\gamma\omega},\qquad
\omega_p^2=\frac{ne^2}{\epsilon_0m}.}
$$
The paper's $\epsilon$ is this dimensionless <relative permittivity>, rather than the absolute <permittivity> $\epsilon_0\epsilon_r$. Independent classes of bound <Electrons> contribute additive susceptibilities, usually with different <resonance> frequencies and <optical oscillator strengths>.
Writing $D_\omega=(\omega_0^2-\omega^2)^2+\gamma^2\omega^2$ separates the response into
$$
\epsilon_r'=1+\frac{\omega_p^2(\omega_0^2-\omega^2)}{D_\omega},\qquad
\epsilon_r''=\frac{\omega_p^2\gamma\omega}{D_\omega}.
$$
At zero <frequency> the <real part> is $1+\omega_p^2/\omega_0^2$ and the <imaginary part> vanishes. Near <resonance> the <imaginary part> has a positive <photon absorption> peak and the <real part> changes rapidly, passing through $1$ at $\omega_0$. Above <resonance> the <real part> can become negative when <damping> is sufficiently weak; at very high <frequency> it approaches $1$ from below while the <imaginary part> tends to zero. Finite <damping> removes the undamped divergence.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2012-lorentz-optics.png]
{title=Real and imaginary relative permittivity, optical <reflectance> and attenuation in a <Lorentz oscillator model>}
For a nonmagnetic material take the passive square-root branch $N=\eta+i\kappa=\sqrt{\epsilon_r}$, with $\kappa\geq0$. A <plane wave> then has an intensity attenuation coefficient $\alpha=2\omega\kappa/c$. Thus the imaginary <permittivity> expresses dissipation, whereas the actual <photon absorption> coefficient is determined by the imaginary <refractive index>. The normal-incidence <power> <reflectance> is
$$
R=\frac{(\eta-1)^2+\kappa^2}{(\eta+1)^2+\kappa^2}.
$$
A large refractive-index mismatch produces <wave reflection> even without loss. In the undamped model $\epsilon_r<0$ between $\omega_0$ and $\sqrt{\omega_0^2+\omega_p^2}$, so the field is evanescent and <wave reflection> is total in a semi-infinite sample. With <damping> the response includes both <wave reflection> and <photon absorption>; a peak in the imaginary <permittivity> should not simply be identified with a <reflectance> peak.
To infer the short-wavelength background from $R=0.35$, assume weak <photon absorption> and take the physical <semiconductor> branch $\eta>1$. Then
$$
\sqrt R=\frac{\eta-1}{\eta+1},\qquad
\eta=\frac{1+\sqrt{0.35}}{1-\sqrt{0.35}}=3.8973,
$$
so
$$
\boxed{\epsilon_b\simeq\eta^2=15.19\simeq15.}
$$
This is an estimate of the valence-electron background in a <frequency> window above the doped-carrier response and below the effective bound-electron <resonance>. One <reflectance> value alone does not determine both parts of a complex dielectric constant. Even for real $\epsilon_r$, a reciprocal-index branch gives the same $R$; the ordinary background index greater than unity selects the branch used here.
Using the supplied conventional cubic <lattice> parameter, the participating valence-electron <number density> is
$$
n_v=\frac{4\times4}{(0.56\times10^{-9}\,\mathrm m)^3}
=9.11\times10^{28}\,\mathrm{m^{-3}}.
$$
Using $m=m_e$ for the stipulated free-electron comparison gives
$$
\omega_{pv}=\sqrt{\frac{n_ve^2}{\epsilon_0m_e}}
=1.703\times10^{16}\,\mathrm{s^{-1}},\qquad
\boxed{\hbar\omega_{pv}=11.21\,\mathrm{eV}.}
$$
An <angular frequency> “in eV” means its quantum <energy> $\hbar\omega$, not $h\omega$. No doped-carrier <effective mass> should be inserted into this valence-electron count.
The low-frequency limit of the \i[bound-electron contribution] obeys
$$
\epsilon_b=1+\frac{\omega_{pv}^2}{\omega_0^2}.
$$
Consequently
$$
\boxed{\hbar\omega_0=\frac{11.21\,\mathrm{eV}}{\sqrt{15.19-1}}
=2.98\,\mathrm{eV}\simeq3\,\mathrm{eV}.}
$$
This effective <resonance> is much larger than the quoted $0.2\,\mathrm{eV}$ <band gap>. There is no contradiction: the single oscillator has been assigned the strength of all sixteen participating valence <Electrons> per conventional cell, whereas the <band gap> is only the lowest interband excitation threshold. A real spectrum distributes <optical oscillator strength> over many transitions. Equating the oscillator <resonance> to the lowest threshold would misrepresent the assumed background response. Here “low <frequency>” means low relative to $\omega_0$; the static response of the \i[doped conducting sample] also contains its free carriers.
The additional mobile carriers in doped <indium antimonide> contribute a <Drude frequency response>, obtained by setting their restoring <frequency> to zero. At <infrared> frequencies well below the effective valence <resonance>, the sum is approximately
$$
\epsilon_r(\omega)\simeq\epsilon_b-
\frac{\omega_{pc}^2}{\omega(\omega+i\gamma_c)},\qquad
\omega_{pc}^2=\frac{n_ce^2}{\epsilon_0m_c^*}.
$$
At the shorter wavelengths the carrier term is small and $R$ approaches the background value. As <wavelength> increases, the negative free-carrier contribution lowers the real <permittivity> towards unity. For weak <damping>, refractive-index matching to <vacuum> gives the <Drude-Lorentz reflectivity minimum> at
$$
\boxed{\omega_{\min}\simeq\frac{\omega_{pc}}{\sqrt{\epsilon_b-1}},\qquad
\lambda_{\min}\simeq\frac{2\pi c\sqrt{\epsilon_b-1}}{\omega_{pc}}.}
$$
This minimum is near $\epsilon_r'=1$, not its zero. At the slightly lower screened <plasma frequency> $\omega_{pc}/\sqrt{\epsilon_b}$, the real <permittivity> crosses zero; at longer wavelengths it becomes negative and the material becomes strongly reflective. Finite <damping> prevents a perfectly vanishing minimum and rounds the edge. At still lower <frequency> the large conducting response also drives $R$ towards unity.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2012-doped-reflectance.png]
{title=Illustrative <Drude-Lorentz reflectivity minimum>: increasing carrier density shifts the minimum to shorter wavelengths}
For fixed carrier <effective mass> and background <permittivity>, $\omega_{pc}\propto\sqrt{n_c}$ and $\lambda_{\min}\propto n_c^{-1/2}$. The observed shift of the minimum towards shorter <wavelength> as doping increases is therefore the expected free-carrier trend. The figure is an original model illustration, not a digitization or quantitative fit of the supplied curves; the carrier <effective mass>, <damping> and their density dependence have not been specified.
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