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Fermat's principle
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== Special cases == === 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}}136β8), although in their subsequent discussion of anisotropic crystals, they note that the ray and wave-normal directions generally differ (pp.{{nnbsp}}792β4), 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}}804β5).</ref> === Homogeneous media: rectilinear propagation === In a homogeneous medium (also called a ''uniform'' medium), all the secondary wavefronts that expand from a given primary wavefront {{mvar|W}} in a given time {{math|Ξ''t''}} are [[congruence (geometry)|congruent]] and similarly oriented, so that their envelope {{mvar|W′}} may be considered as the envelope of a ''single'' secondary wavefront which preserves its orientation while its center (source) moves over {{mvar|W}}. If {{mvar|P}} is its center while {{mvar|P′}} is its point of tangency with {{mvar|W′}}, then {{mvar|P′}} moves parallel to {{mvar|P}}, so that the plane tangential to {{mvar|W′}} at {{mvar|P′}} is parallel to the plane tangential to {{mvar|W}} at {{mvar|P}}. Let another (congruent and similarly orientated) secondary wavefront be centered on {{mvar|P′}}, moving with {{mvar|P}}, and let it meet its envelope {{mvar|W″}} at point {{mvar|P″}}. Then, by the same reasoning, the plane tangential to {{mvar|W″}} at {{mvar|P″}} is parallel to the other two planes. Hence, due to the congruence and similar orientations, the ray directions {{mvar|PP′}} and {{mvar|P′P″}} are the same (but not necessarily normal to the wavefronts, since the secondary wavefronts are not necessarily spherical). This construction can be repeated any number of times, giving a straight ray of any length. Thus a homogeneous medium admits rectilinear rays.<ref>[[#deWitte|De Witte, 1959]] (p.{{nnbsp}}295, col.{{nnbsp}}1 and Figure 2), states the result and condenses the explanation into one diagram.</ref>
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