Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2019 / Paper 1 / 9 / b

Choose the mirror as the line $y=0$, the source as $(x_A,h_A)$ and the observer as $(x_B,h_B)$, with positive heights. Within the plane of incidence, let the reflection point be $(s,0)$. In a uniform medium the optical length is a constant index timesStationarity with respect to the point on the mirror givesThese two ratios are the sines of the angles to the normal, with the stationary point between the two projected endpoints. ThereforeThe angles lie in $[0,\pi /2)$, where sine is injective. Varying a second tangential coordinate in three dimensions also enforces that the source, normal and reflected ray lie in one plane. This derives the law of reflection from Fermat principle.

Place the interface at $y=0$, with source $(x_A,h_1)$ and observer $(x_B,-h_2)$ in the two uniform media. Let $(s,0)$ be the crossing point. The optical length isDifferentiating with respect to $s$ and setting the result to zero givesRecognizing the ratios as the sines of the angles measured from the interface normal proves Snell's law:The stationarity condition is the continuity of the tangential optical momentum across the planar interface.