The coefficient of n and n' can be identified as the sines of the incidence and refraction angles i and i' which therefore demonstrates Snell's law. Using the time delay At, instead of the optical path I, and the similar condition d(At )/dy = 0, we obtain the other form (sini')/v' = (sini)/v, where the light velocities v and v' are respectively c/n and c/n'.
A non-homogeneous medium with a refractive index radially distributed from a pole as n = n0/(1-r2/a2), where n0 and a are constants, provides curved rays and leads to an inversion transformation imaging; this has been investigated by Maxwell  and is known as Maxwell's fish-eye. The determination of the propagation curves in a gradient index medium, has recently found several important developments in optical fibers. Given an index distribution, systematic determinations of
Fig. 1.14 Derivation of Snell's law from Fermat's principle
Fig. 1.14 Derivation of Snell's law from Fermat's principle y n the optical path lines which satisfy a stationary value of /nds have been treated by Caratheodory .
Calculus of variations is of considerable importance because it allows deriving theorems which have analogue significance in various fields of physics. There is a close analogy between Fermat's principle in geometrical optics and in dynamics with the movement of a particle submitted to a force derived from a potential U (x, y, z). Hamilton demonstrated that the action A may be expressed by J\JU + a di - where a is a constant -, and should be minimal or at least stationary. Hamilton's principle of stationary action and his masterful comparison with Fermat's principle became of great value in De Broglie's wave mechanics (1924).
Considering a source point in a homogeneous medium, the emitted rays are all normal to the propagating pencil wavefronts which are concentric spherical surfaces, also said to be homocentric wavefronts. The rays are all orthogonal to these surfaces and then form a normal congruence. If the rays are now refracted by an optical surface in a second homogeneous medium, then each propagation line remains straight; such a congruence is called a normal rectilinear congruence. Malus showed in 1808  that, whatever the surface shape, the refracted or reflected pencil - which generally is no longer homocentric - again forms a normal rectilinear congruence. Dupin (1816), Quetelet (1825) and others generalized this property which is known as Malus-Dupin theorem:
^ In any homogeneous medium rays propagate in normal rectilinear congruence.
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