# How does Fermat's principle of least time follow from Huygen's principle?

When I was first introduced to Snell's Law, I was shown a derivation using Fermat's principle of least time. Using this same principle you can show that the angle of incidence must be equal to the angle of reflection. I was later again shown this, but instead of using Fermat's principle, all of this was derived using Huygen's principle. I think I've read somewhere that Fermat's principle can be mathematically derived from Huygen's principle. How can this be done?

## 2 Answers

The only way of doing so that I can think of is demonstrate that Huygen's Principle and Fermat's Principle are both equivalent to the eikonal equation. This involves the calculus of variations. I haven't studied optics in a pretty long time, so maybe there's a simpler derivation that I am unaware of or have forgotten. Chapter 10 of these lecture notes provides a step by step derivation of the equivalence of Huygen's Principle, Fermat's Principle, and the eikonal equation.

• Minor comment to the post (v1): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. – Qmechanic Sep 3 '18 at 5:14

Thomas Young thought the connection was obvious. "The principle of Fermat," he declared, "although it was assumed by that mathematician on hypothetical, or even imaginary grounds, is in fact a fundamental law with respect to undulatory motion, and is explicitly the basis of every determination in the Huygenian theory" (Misc. Works, vol. 1, pp. 225–6).

Because I'm a bit slow by Young's standards, I wouldn't have said "explicitly"; but I have attempted an elementary explanation in a partial draft of Wave Foundations of Ray Optics (free download), pp. 1–9.

Abstract of Chapter 1: "The Huygens-Fermat principle"

Huygens implicitly defined a ray as a path by which light can travel between successive positions of a wavefront in the least time, assuming this time to be the same as if the light were emitted by a point-source on the earlier wavefront. Conversely, Fermat's principle assumes that the propagation time between any two points on a path is the same as if the first point were a secondary source. The equivalence of Huygens' and Fermat's approaches, which was recognized by Laplace and by Thomas Young, is here explained from first principles. Huygens' assumption is equivalent to another: that the normal velocity of a wavefront does not depend on its curvature or intensity. Ironically, the actual generation of secondary waves is not needed in Huygens' system except for reflection, and then only at the reflecting surface (provided that his implicit definition and supporting assumption are taken as fundamental). The foregoing principles lead to rectilinear propagation in uniform media, the coincidence of rays and wave-normals in isotropic media, the relation between the ray velocity and the wave-normal velocity, and the general and ordinary laws of refraction and reflection. In due course, these laws are restated in terms of wave slowness and refractive index. Conservation of the tangential component of wave slowness (Hamilton's "common perpendicular" rule) is demonstrated from wavefront continuity, and yields a simple retrospective explanation of early geometric formulations of the ordinary law of refraction. The theory of reciprocal surfaces is developed in less than three pages, showing how the wave-slowness surfaces (or the index surfaces) offer a simpler alternative to Huygens' construction.