  We numerically study the effect of the quantum spill-out (QSO) on the plasmon
mode indices of an ultra-thin metallic slab, using the Fourier modal method
(FMM). To improve the convergence of the FMM results, a novel nonlinear
coordinate transformation is suggested and employed. Furthermore, we present a
perturbative approach for incorporating the effects of QSO on the plasmon mode
indices, which agrees very well with the full numerical results. The
perturbative approach also provides additional physical insight, and is used to
derive analytical expressions for the mode indices using a simple model for the
dielectric function. The analytical expressions reproduce the results obtained
from the numerically-challenging spill-out problem with much less effort and
may be used for understanding the effects of QSO on other plasmonic structures.
