  The detailed optical properties have been determined for the iron-based
materials $A$Fe$_2$As$_2$, where $A=\,$Ca, Sr, and Ba, for light polarized in
the iron-arsenic ($a-b$) planes over a wide frequency range, above and below
the magnetic and structural transitions at $T_N =$ 172, 195, and 138 K,
respectively. The real and imaginary parts of the complex conductivity are fit
simultaneously using two Drude terms in combination with a series of
oscillators. Above $T_N$, the free-carrier response consists of a weak, narrow
Drude term, and a strong, broad Drude term, both of which show only a weak
temperature dependence. Below $T_N$ there is a slight decrease of the plasma
frequency but a dramatic drop in the scattering rate for the narrow Drude term,
and for the broad Drude term there is a significant decrease in the plasma
frequency, while the decrease in the scattering rate, albeit significant, is
not as severe. The small values observed for the scattering rates for the
narrow Drude term for $T\ll{T_N}$ may be related to the Dirac cone-like
dispersion of the electronic bands. Below $T_N$ new features emerge in the
optical conductivity that are associated with the reconstruction Fermi surface
and the gapping of bands at $\Delta_1 \simeq$ 45 $-$ 80 meV, and $\Delta_2
\simeq$ 110 $-$ 210 meV. The reduction in the spectral weight associated with
the free carriers is captured by the gap structure, specifically, the spectral
weight from the narrow Drude term appears to be transferred into the low-energy
gap feature, while the missing weight from the broad term shifts to the
high-energy gap.
