 Solution

Fermat principle states that the optical path length $\int nds$, equivalently the travel time in the nondispersive stationary isotropic ray model, is stationary for the actual light ray between fixed endpoints. It need not be a minimum in every optical system.

For two homogeneous media separated by $y=0$, place the fixed endpoints at $(0,h_1)$ and $(D,-h_2)$, with positive heights. If the ray crosses at $(x,0)$, its optical path isStationarity with respect to the crossing point givesThe two fractions are $\sin {\theta }_1$ and $\sin {\theta }_2$, with angles measured from the interface normal. Thuswhich is Snell's law. Dividing $\mathcal{L}$ by the vacuum light speed gives the corresponding travel time; minimizing geometric distance without refractive-index weighting would incorrectly predict a straight ray through unlike media.