  We have investigated the electronic and optical properties of
(Sr$_{1-x}$Ca$_{x}$)$_2$IrO$_4$ (x= 0 - 0.375) and
(Sr$_{1-y}$Ba$_y$)$_2$IrO$_4$ (y= 0 - 0.375) epitaxial thin-films, in which the
bandwidth is systematically tuned via chemical substitutions of Sr ions by Ca
and Ba. Transport measurements indicate that the thin-film series exhibits
insulating behavior, similar to the J$_{eff}$= 1/2 spin-orbit Mott insulator
Sr$_2$IrO$_4$. As the average A-site ionic radius increases from
(Sr$_{1-x}$Ca$_{x}$)$_2$IrO$_4$ to (Sr$_{1-y}$Ba$_y$)$_2$IrO$_4$, optical
conductivity spectra in the near-infrared region shift to lower energies, which
cannot be explained by the simple picture of well-separated J$_{eff}$= 1/2 and
J$_{eff}$= 3/2 bands. We suggest that the two-peak-like optical conductivity
spectra of the layered iridates originates from the overlap between the
optically-forbidden spin-orbit exciton and the inter-site optical transitions
within the J$_{eff}$= 1/2 band. Our experimental results are consistent with
this interpretation as implemented by a multi-orbital Hubbard model
calculation: namely, incorporating a strong Fano-like coupling between the
spin-orbit exciton and inter-site d-d transitions within the J$_{eff}$= 1/2
band.
