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 $Tll{T_N}$ 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.
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