Analytical ray tracing in GRIN lenses: variational and series approaches

We present two analytical methods for describing light propagation and related characteristics-such as optical power and optical path length-in gradient refractive index (GRIN) lenses of arbitrary geometry. The first, a direct variational technique based on Fermat's principle, yields analytical expressions for light paths in inhomogeneous media. The second, a Taylor series approach adapted to boundary-value problems, provides complementary analytical insight. Both methods enable ray tracing through complete optical systems, including GRIN lenses embedded in media of different refractive indices, with interface conditions imposed via Snell's law. Comparison with numerical solutions of the Euler-Lagrange equations shows excellent agreement, confirming the accuracy of our approach. The results are directly applicable to modeling the human crystalline lens and other biological or engineered GRIN systems.