Fermat's Principle

Fermat's principle, also called the principle of least time, states that a light ray between fixed endpoints makes its optical path length stationary. For a curve , the optical path length is the functional

(1)

where is the index of refraction and is the arc length element. In a stationary, isotropic, nondispersive medium the ray speed is , where is the speed of light in vacuum. The travel time is then , so the two functionals have the same stationary paths. If is constant, is proportional to arc length and the ray is a straight line (Open University 2016, pp. 28-30).

Consider two uniform media separated by the x-axis, with indices of refraction and . The two segments from fixed points and to the crossing point give

(2)

where and . For , differentiating and setting gives

(3)

where and are the angles the two segments make with the normal to the interface. This is Snell's law of refraction, which Fermat derived using this principle (Andersen 1983).