Past exam of the physics 2 paper of the University of Oxford / 2017 resit / 9 / a / Solution

Fermat's principle states that the travel time, or equivalently the optical path length $\int nds$ in a stationary medium, is stationary under small variations of the path with endpoints held fixed. It need not be a global minimum.

Let a light ray join a point at height $h_1$ above a plane interface to a point at depth $h_2$ below it, with horizontal separation $D$. If the crossing point is horizontally $\xi$ from the first endpoint, its optical path length isSetting its derivative to zero givesThe two geometric ratios are the sines of the incident and refracted angles, measured from the interface normal vector. Hence Snell's law is