Fast Hessian-free finite element ray tracing method for light transport in gradient-index media

Accurate modeling of light transport in graded refractive index (GRIN) media is critical for applications such as radiotherapy, vat photopolymerization, and photolithography. Finite element ray tracing (FE-RT) can provide accurate light transport simulations for inhomogeneous refractive index fields by discretizing the medium into finite elements, within which the refractive index is interpolated. Existing methods are limited to higher-order elements (p ≥ 2) due to their reliance on the Hessian of the shape functions. We introduce a new FE-RT method with a mixed state-space formulation that does not require the Hessian. This generalizes FE-RT to elements of order p ≥ 1 and improves efficiency. The framework captures key optical phenomena, integrates with multi-physics simulations, and is optimized for performance using GPU-parallelized algorithms. Analytical validation confirms convergence to closed-form ray paths, achieving near-machine-precision numerical accuracy. Performance tests show strong scalability: for a refinement from 640 to 300k elements, the runtime only increased from 40 to 100 ms. A fiber optic damage case study illustrates the ability to model GRIN effects, rough surface scattering, and coupled simulations. These advances extend FE-RT's efficiency and applicability, enabling real-time simulation, design optimization, and predictive modeling in complex optical systems.