Calculation of the Luneberg optical lens and applications

Abstract

This paper presents a numerical approach for designing radially symmetric Luneburg lenses with arbitrary source and image positions. The method reconstructs the refractive index distribution by inverting an Abel-type relation and uses adaptive Gauss–Kronrod quadrature to ensure high accuracy without manual tuning. The algorithm is validated against the classical Luneburg profile and applied to two focusing scenarios: transformation of a plane wave into a converging wave with a prescribed focal distance, and focusing of radiation from a finite-distance point source to a target point. Ray tracing with an explicit Euler scheme and Huygens-based wavefront simulations confirm that an initially planar front segment collapses precisely to the designed focus. Using the optical–hydrodynamic analogy, the obtained refractive index distribution is converted into a seamount depth profile, and nonlinear shallow-water simulations demonstrate significant wave amplitude amplification near the predicted focal area. The results provide a universal framework for the synthesis of gradient-index lenses and offer practical insights into tsunami wave focusing caused by natural seabed topography.