|∇S|² = n²(r) — rays follow the gradient of the phase, bending in gradient-index media
GRIN Profile
Eikonal equation: In geometrical optics, wavefronts S satisfy |∇S|² = n²(r). Rays are the trajectories perpendicular to wavefronts, following d/ds(n dr/ds) = ∇n. Gradient-index (GRIN) media continuously bend rays. Radial profiles focus parallel rays like a lens. Nature uses GRIN optics in fish eyes and optical fibers.