We measured the THz reflectance properties of a high quality epitaxial thin film of the Fe-based superconductor BaFe$_{1.84}$Co$_{0.16}$As$_2$ with T$_c$=22.5 K. The film was grown by pulsed laser deposition on a DyScO$_3$ substrate with an epitaxial SrTiO$_3$ intermediate layer. The measured $R_S/R_N$ spectrum, i.e. the reflectivity ratio between the superconducting and normal state reflectance, provides clear evidence of a superconducting gap $Delta_A$ close to 15 cm$^{-1}$. A detailed data analysis shows that a two-band, two-gap model is absolutely necessary to obtain a good description of the measured $R_S/R_N$ spectrum. The low-energy $Delta_A$ gap results to be well determined ($Delta_A$=15.5$pm$0.5 cm$^{-1}$), while the value of the high-energy gap $Delta_B$ is more uncertain ($Delta_B$=55$pm$7 cm$^{-1}$). Our results provide evidence of a nodeless isotropic double-gap scenario, with the presence of two optical gaps corresponding to 2$Delta/kT_c$ values close to 2 and 7.
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