  The electronic structures of the heavy-fermion superconductor CeRh$_2$As$_2$
with the local inversion symmetry breaking and the reference material
LaRh$_2$As$_2$ have been investigated by using experimental optical
conductivity ($\sigma_1(\omega)$) spectra and first-principal DFT calculations.
In the low-temperature $\sigma_1(\omega)$ spectra of CeRh$_2$As$_2$, a
$4f$-conduction electron hybridization and heavy quasiparticles are clearly
indicated by a mid-infrared peak and a narrow Drude peak. In LaRh$_2$As$_2$,
these features are absent in the $\sigma_1(\omega)$ spectrum, however, it can
nicely be reproduced by DFT calculations. For both compounds, the combination
between a local inversion symmetry breaking and a large spin-orbit (SO)
interaction plays an important role for the electronic structure, however, the
SO splitting bands could not be resolved in the $\sigma_1(\omega)$ spectra due
to the small SO splitting size.
