  With infrared spectroscopy we studied the out-of-plane ($c$-axis) response of
the iron arsenide superconductor ACa$_2$Fe$_4$As$_4$F$_2$ (A = Rb,Cs) which has
a bilayer structure similar to the high $T_c$ cuprates YBa$_2$Cu$_3$O$_7$
(YBCO) and Bi$_2$Sr$_2$CaCu$_2$O$_8$ (Bi2212). In analogy to the cuprates, we
observe a superconductivity-induced transverse plasma mode (tPM) and a phonon
anomaly that are both signatures of local electric field effects that arise
from a large difference between the local conductivities in the intra- and
inter-bilayer regions. Using a multilayer model developed for the cuprates, we
obtain a good description of the $c$-axis response and derive the local
conductivities at $T \simeq T_c$ of $\sigma_1^{\mathrm{bl}}(\omega \rightarrow
0) \simeq$ 1\,000 $\Omega^{-1}\mathrm{cm}^{-1}$ and
$\sigma_1^{\mathrm{int}}(\omega \rightarrow 0) \simeq$ 15
$\Omega^{-1}\mathrm{cm}^{-1}$, respectively, that are similar to the ones
previously found in underdoped YBCO. Different from the cuprates, we find no
evidence of a normal state pseudogap in terms of a partial suppression of the
low-energy electronic states that sets in already well above $T_c$. There is
also no clear sign of an onset of precursor superconducting pairing
correlations well above $T_c \simeq$ 30~K. This highlights that the pseudogap
and the precursor superconducting pairing well above $T_c$ are unique features
of the cuprates with their strong electronic correlations and, for example, not
just the result of a strongly anisotropic electronic response due to the
layered crystal structure.
