  We have investigated optical conductivity spectra of $n$- and $p$-type
Ba$_8$Ga $_{16}$Sn$_{30}$ ($\alpha$-BGS) with type-VIII clathrate structure, at
temperatures from 296\,K down to 6\,K with a terahertz time-domain spectrometer
(0.2\,-\,2.5\,THz). The continuous spectra contributed from charge carriers are
dispersive in this frequency range and also temperature- and carrier
type-dependent. The Drude-Smith model taking multiple-scatterings of charge
carriers into account well reproduces those data. The relaxation rate of the
$n$-type carriers decreases more sharply than that in the $p$-type material,
suggesting that a stronger electron-phonon interaction may exist in the
$n$-type than in the $p$-type. On the other hand, the localized infrared-active
modes observed at 1.3\,THz and 1.7\,THz, identified as the rattling phonons of
the Ba$^{2+}$ ion's quasi-on-center vibrations, become soft and broad
significantly with decreasing temperature as well as observed in type-I BGS and
BGG (Ba$_8$Ga$_{16}$Ge$_{30}$) clathrates. The softening in the $n$-type is
smaller by about 30% than in the $p$-type, whereas the linewidth brodening is
almost the same independently on the carrier type. The difference in the
softening is discussed with a scenario where the interaction of rattling
phonons with carriers can modify the anharmonic potential of the guest ions.
The anomalous broadening at low temepratures is also discussed by the
impurity-scattering model presented for a rattling-phonon system strongly
hybridized with acoustic cage phonons.
