  We investigated a high-quality MgB$_{2}$ thin film with a thickness of
$\sim$1000 nm on an Al$_{2}$O$_{3}$ substrate using optical spectroscopy. We
measured the reflectance spectra of the film at various temperatures both
below, and above, the superconducting transition temperature, $T_c$ $\simeq$ 40
K. An earlier study showed that when the sample surface is exposed to air the
optical properties of the surface change immediately, however, the saturated
change is negligibly small in the far-infrared region. The optical conductivity
spectrum in the normal state shows two (narrow and broad) Drude modes, with the
narrow Drude mode being dominant in the low frequency region below 1000
cm$^{-1}$. Our study, which uses a good-quality sample, provides more reliable
data on the optical properties of MgB$_2$, in a similar spectral range. The
optical data is analyzed further using an extended Drude model, and the
electron-phonon spectral density function, $\alpha^2F(\omega)$, is extracted.
The spectral density function $\alpha^2F(\omega)$ features two peaks: a small
one near 114 cm$^{-1}$, and a strong peak around the 550 cm$^{-1}$ where the
B-B bond stretching phonon exists. Our data in the superconducting state does
not show the expected energy shift of the onset of scattering associated with
the $\alpha^2F(\omega)$ peaks.
