  Angle-resolved photoemission spectroscopy (ARPES) presents significant
simplications in analyzing strictly two-dimensional (2D) materials, but even
the most anisotropic physical systems display some residual
three-dimensionality. Here we demonstrate how this third dimension manifests
itself in ARPES spectra of quasi-2D materials by considering the example of the
cuprate Bi$_2$Sr$_2$CaCu$_2$O$_{8}$ (Bi2212). The intercell, interlayer
hopping, which is responsible for $k_z$-dispersion of the bands, is found to
induce an irreducible broadening to the ARPES lineshapes with a characteristic
dependence on the in-plane momentum $k_\parallel$. Our study suggests that
ARPES lineshapes can provide a direct spectroscopic window for establishing the
existence of coherent c-axis conductivity in a material via the detection of
this new broadening mechanism, and bears on the understanding of 2D to 3D
crossover and pseudogap and stripe physics in novel materials through ARPES
experiments.
