  We design an all-dielectric L\"uneburg lens as an adiabatic space-variant
lattice explicitly accounting for finite film thickness. We describe an
all-analytical approach to compensate for the finite height of subwavelength
dielectric structures in the pass-band regime. This method calculates the
effective refractive index of the infinite-height lattice from effective medium
theory, then embeds a medium of the same effective index into a slab waveguide
of finite height and uses the waveguide dispersion diagram to calculate a new
effective index. The results are compared with the conventional numerical
treatment - a direct band diagram calculation, using a modified
three-dimensional lattice with the superstrate and substrate included in the
cell geometry. We show that the analytical results are in good agreement with
the numerical ones, and the performance of the thin-film L\"uneburg lens is
quite different than the estimates obtained assuming infinite height.
