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

Question

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?

Answer 1

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.

Answer 2

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 (doi.org/10.5281/zenodo.3901935), sections 1.2 and 1.3.

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

Huygens' principle, too often asserted independently, is here developed with Huygens' original inevitability and a modern generality. Fermat's principle, too often stated in isolation and without even specifying the applicable propagation speed, is here developed as a consequence of Huygens' principle and justified in terms of other notions of a ray (line of sight, narrow beam). The equivalence of Huygens' construction and Fermat's principle — recognized by Young, Laplace, and Fresnel, further developed by Lorentz, proven analytically for two dimensions by De Witte, but neglected in textbooks — is here demonstrated in its full generality by elementary geometric arguments. The foregoing principles lead to rectilinear propagation in homogeneous media, the coincidence of rays and wave-normals in isotropic media, the general relation between the ray velocity and the normal velocity, and the general and ordinary laws of refraction and reflection. The laws of refraction and reflection are restated as conservation of the tangential component of wave slowness, leading to Hamilton's "common perpendicular" rule. The other feature of Hamilton's construction — the relation between the wave-slowness surface and the direction and magnitude of the ray slowness — is derived in two pages, and the construction is then compared with that of Huygens. For isotropic media, Hamilton's construction yields an ironic retrospective explanation of early geometric formulations of the ordinary law of refraction. In the last section, the refractive index is introduced as a normalized wave slowness, and previous results are restated in terms of refractive indices. Historical notes, with sources, are included.