  We study the optical properties of the Ruddlesden-Popper series of iridates
Sr$_{n+1}$Ir$_n$O$_{3n+1}$ ($n$=1, 2 and $\infty$) by solving the
Bethe-Salpeter equation (BSE), where the quasiparticle (QP) energies and
screened interactions $W$ are obtained by the $GW$ approximation including
spin-orbit coupling. The computed optical conductivity spectra show strong
excitonic effects and reproduce very well the experimentally observed
double-peak structure, in particular for the spin-orbital Mott insulators
Sr$_2$IrO$_4$ and Sr$_3$Ir$_2$O$_7$. However, $GW$ does not account well for
the correlated metallic state of SrIrO$_3$ owing to a much too small band
renormalization, and this affects the overall quality of the optical
conductivity. Our analysis describes well the progressive redshift of the main
optical peaks as a function of dimensionality ($n$), which is correlated with
the gradual decrease of the electronic correlation (quantified by the
constrained random phase approximation) towards the metallic $n=\infty$ limit.
We have also assessed the quality of a computationally cheaper BSE approach
that is based on a model dielectric function and conducted on top of DFT+$U$
one-electron energies. Unfortunately, this model BSE approach does not
accurately reproduce the outcome of the full $GW$+BSE method and leads to
larger deviations to the measured spectra.
