  High resolution angle-resolved photoemission spectroscopy (ARPES)
measurements are carried out on CaKFe$_4$As$_4$, KCa$_2$Fe$_4$As$_4$F$_2$ and
(Ba$_{0.6}$K$_{0.4}$)Fe$_2$As$_2$ superconductors. Clear evidence of band
folding between the Brillouin zone center and corners with a ($\pi$,$\pi$) wave
vector has been found from the measured Fermi surface and band structures in
all the three kinds of superconductors. A dominant $\sqrt{2}\,\times\,\sqrt{2}$
surface reconstruction is observed on the cleaved surface of CaKFe$_4$As$_4$ by
scanning tunneling microscopy (STM) measurements. We propose that the commonly
observed $\sqrt{2}\,\times\,\sqrt{2}$ reconstruction in the FeAs-based
superconductors provides a general scenario to understand the origin of the
($\pi$,$\pi$) band folding. Our observations provide new insights in
understanding the electronic structure and superconductivity mechanism in
iron-based superconductors.
