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The optical properties of the new iron-based superconductorCsCa$_2$Fe$_4$As$_4$F$_2$ with $T_c \sim 29$~K have been determined. In thenormal state a good description of the low-frequency response is obtained witha superposition of two Drude components of which one has a very low scatteringrate (narrow Drude-peak) and the other a rather large one (broad Drude-peak).Well below $T_c \sim 29$~K, a pronounced gap feature is observed which involvesa complete suppression of the optical conductivity below $\sim$ 110~cm$^{-1}$and thus is characteristic of a nodeless superconducting state. The opticalresponse of the broad Drude-component can be described with a dirty-limitMattis-Bardeen-type response with a single isotropic gap of $2\Delta \simeq14$~meV. To the contrary, the response of the narrow Drude-component is in theultra-clean-limit and its entire spectral weight is transferred to thezero-frequency $\delta(\omega)$ function that accounts for the loss-freeresponse of the condensate. These observations provide clear evidence for aband-selective coexistence of clean- and dirty-limit superconductivity withnodeless gaps in CsCa$_2$Fe$_4$As$_4$F$_2$.

Band-selective clean- and dirty-limit superconductivity with nodeless gaps in the bilayer iron-based superconductor CsCa$_2$Fe$_4$As$_4$F$_2$