  We performed an optical spectroscopy measurement on single crystals of
$\mathrm{Ba_2Ti_2Fe_2As_4O}$, which is a newly discovered superconductor
showing a coexistence of superconductivity and density wave order. The study
reveals a spectral change related to the formation density wave energy gap
below $T_{DW}\approx$125 K, leading to the removal of about half of the
multiple Fermi surface sheets. The ratio of 2$\Delta_{DW}$/$k_B T_{DW}\approx$
11.9 is considerably larger than the mean-field value based on the
weak-coupling BCS theory. At the lowest temperature in the superconducting
state, we observed opening of superconducting energy gaps $\Delta_1(0) =3.4$
meV and $\Delta_2(0)=7.9$ meV. The properties of superconducting state in
$\mathrm{Ba_2Ti_2Fe_2As_4O}$ are similar to that in
$\mathrm{BaFe_{1.85}Co_{0.15}As_2}$.
