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

TL;DR

This paper presents an analytical method to design a thin-film dielectric Luneburg lens by accounting for finite thickness effects, providing accurate effective index calculations that improve upon traditional numerical approaches.

Contribution

An all-analytical approach for designing thin-film dielectric Luneburg lenses that accurately compensates for finite height effects using effective medium and waveguide theories.

Findings

Analytical results agree well with numerical simulations.

Finite thickness significantly affects lens performance.

The method improves design accuracy over conventional models.

Abstract

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.