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We design an all-dielectric L\"uneburg lens as an adiabatic space-variantlattice explicitly accounting for finite film thickness. We describe anall-analytical approach to compensate for the finite height of subwavelengthdielectric structures in the pass-band regime. This method calculates theeffective refractive index of the infinite-height lattice from effective mediumtheory, then embeds a medium of the same effective index into a slab waveguideof finite height and uses the waveguide dispersion diagram to calculate a neweffective index. The results are compared with the conventional numericaltreatment - a direct band diagram calculation, using a modifiedthree-dimensional lattice with the superstrate and substrate included in thecell geometry. We show that the analytical results are in good agreement withthe numerical ones, and the performance of the thin-film L\"uneburg lens isquite different than the estimates obtained assuming infinite height.

Design of thin-film photonic metamaterial L\"uneburg lens using analytical approach