Ray tracing in gradient-index media: a comparative analysis of discretization-based methods

The calculation of ray trajectories through gradient-index (GRIN) media by solving the ray equation often represents a significant challenge. While closed-form analytical solutions exist for certain specific refractive index distributions, most practical cases require numerical methods. To address this, various approaches have been developed that avoid directly solving the ray equation by discretizing either the refractive medium, the geometric path length, or the optical path length, reconstructing the ray trajectory by iteratively applying Snell's law. In this paper, three numerical ray-tracing methods based on these distinct discretization schemes are presented: medium discretization, geometric path length discretization, and optical path length discretization. A comparative analysis is performed using the Luneburg lens, for which an exact analytical solution exists, demonstrating that optical path length discretization consistently achieves superior accuracy and provides a robust framework for ray tracing in GRIN media. A thorough evaluation of the accuracy and limitations of each method is provided; additionally, the advantages of the optical path length discretization approach over the well-known Runge-Kutta scheme are discussed.