In order to determine the orbital characters on the various Fermi surface pockets of the Fe-based superconductors Ba$_{0.6}$K$_{0.4}$Fe$_{2}$As$_{2}$ and FeSe$_{0.45}$Te$_{0.55}$, we introduce a method to calculate photoemission matrix elements. We compare our simulations to experimental data obtained with various experimental configurations of beam orientation and light polarization. We show that the photoemission intensity patterns revealed from angle-resolved photoemission spectroscopy measurements of Fermi surface mappings and energy-momentum plots along high-symmetry lines exhibit asymmetries carrying precious information on the nature of the states probed, information that is destroyed after the data symmetrization process often performed in the analysis of angle-resolved photoemission spectroscopy data. Our simulations are consistent with Fermi surfaces originating mainly from the $d_{xy}$, $d_{xz}$ and $d_{yz}$ orbitals in these materials. ; Comment: 16 pages, 9 figures. Figures modified, typos corrected, appendix added