Gradient-index optics

Remarks

The ray equation is the Euler–Lagrange equation of the optical path length $\int n(r(s)) \, ds$, so the rays are the curves of stationary optical length, as Fermat's principle says. Equivalently, the rays are the geodesics of the Riemannian metric $n(x)^2 \abs{dx}^2$ on $\Omega$, and the optical path length from a point solves the eikonal equation $\abs{\nabla L} = n$. Example: in a slab with $n(x, y, z) = n_0 \sqrt{1 - \alpha^2 x^2}$, rays near the $z$-axis oscillate about it like sine curves, which is how a graded-index fibre guides light. Non-example: if $n$ is constant, the ray equation gives $d^2 r / ds^2 = 0$, so the rays are straight lines; a homogeneous medium is not a gradient-index medium.