Pseudogap and Strong Pairing Induced by Incipient and Shallow Bands in the Quasi-Two-Dimensional KCa$_{2}$Fe$_{4}$As$_{4}$F$_{2}$

The optical properties of KCa$_{2}$Fe$_{4}$As$_{4}$F$_{2}$ (K12442, $T_c = 33.5$~K) and KCa$_{2}$(Fe$_{0.95}$Ni$_{0.05}$)$_{4}$As$_{4}$F$_{2}$ (Ni-K12442, $T_c = 29$~K) have been examined at a large number of temperatures. For both samples, a nodeless superconducting gap is clearly observed in the optical conductivity at 5~K. The superconducting gap $\Delta \simeq 8.7$~meV ($2\Delta/k_{\text{B}}T_{c} \simeq 6.03$) in K12442, pointing towards strong-coupling Cooper pairs, but in sharp contrast, $\Delta \simeq 4.6$~meV ($2\Delta/k_{\text{B}}T_{c} \simeq 3.68$) in Ni-K12442, which agrees with the BCS weak-coupling pairing state. More intriguingly, below $T^{\ast} \simeq 75$~K, the optical conductivity of K12442 reveals a pseudogap that smoothly evolves into the superconducting gap below $T_{c}$, while no such behavior is detected in the electron-doped Ni-K12442. The comparison between the two samples hints that the pseudogap and strong-coupling Cooper pairs in K12442 may be intimately related to the shallow and incipient bands.