Editing Fermat's principle (section)

=== Isotropic media: rays normal to wavefronts ===

In an isotropic medium, because the propagation speed is independent of direction, the secondary wavefronts that expand from points on a primary wavefront in a given ''infinitesimal'' time are spherical,<ref name="De Witte, 1959" /> so that their radii are normal to their common tangent surface at the points of tangency. But their radii mark the ray directions, and their common tangent surface is a general wavefront. Thus the rays are normal (orthogonal) to the wavefronts.<ref>[[#deWitte|De Witte, 1959]], p.{{nnbsp}}295, col.{{nnbsp}}1.</ref>

Because much of the teaching of optics concentrates on isotropic media, treating anisotropic media as an optional topic, the assumption that the rays are normal to the wavefronts can become so pervasive that even Fermat's principle is explained under that assumption, although in fact Fermat's principle is more general.<ref>Even Born & Wolf prove Fermat's principle for the case in which the rays are normal to the wavefronts ([[#BW|2002]], pp.{{nnbsp}}136–8), although in their subsequent discussion of anisotropic crystals, they note that the ray and wave-normal directions generally differ (pp.{{nnbsp}}792–4), and that for a given wave-normal direction, the ray direction is such that the speed of the intersection between the ray-line and the plane wavefront is stationary with respect to variations of the wave-normal direction (pp.{{nnbsp}}804–5).</ref>