Numerical Solution of the Eikonal and Transport Equations for GRIN Lens Antennas by Using the Lax-Friedrichs Sweeping Method

Geometrical Optics (GO) analysis of Graded-Index (GRIN) lenses involves tracing curved rays through inhomogeneous media and solving transport equations to determine the field distribution at the lens output. While efficient GO ray-tracing algorithms are quite fast, they are complex to implement and suffer from caustic problems. Moreover, GO lacks spatial continuity and predefined spatial meshes, making it unsuitable for automatic post-processing in optimization schemes. In this paper, we introduce a novel algorithm for analyzing inhomogeneous lenses by numerically solving the Eikonal and transport equations on a computational grid, bypassing the need for ray tracing. This approach utilizes a numerical technique known as the sweeping method, which iteratively updates the solution to the governing equations across the grid. The sweeping process involves traversing the grid in alternating directions, ensuring convergence by considering boundary conditions and neighboring grid points through Gauss-Seidel iterations. The algorithm presented here is the Lax-Friedrichs Sweeping Method (LFSM), notable for its innovative approach to solving the factored Eikonal and transport equations. This method enables the precise computation of the Eikonal Laplacian term, thereby enhancing the accuracy of the amplitude calculation. Its implementation surpasses traditional ray-tracing in speed, accuracy, computational efficiency, and robustness. For GRIN lens analysis using LFSM, the required inputs are a computational grid, a refractive index map, and the source type and location. This paper details the algorithm’s implementation and validates its effectiveness through analytical benchmarks and practical examples, including a telescopic lens antenna and a spaceborne weather radar lens antenna.