Fermats Principle

In 1657, in a letter to Cureau [73], Pierre de Fermat stated that the law of refraction might be deduced from a minimum principle, and this similarly as the remark by Heron of Alexandria (100 AD) for the reflection on a plane. In 1662, considering optical paths between two points in different media, he proved that a ray follows the optical path which takes the least time since this also entails the direct recovering of Snell's law.

In all generality, Fermat's principle of least time may be enounced as: the time delay for the light to travel from a point A to a point B, through a medium of refractive index n(x, y, z), is minimal or stationary. The ray follows a curvilinear line of elementary length ds, and the local velocity of the light v is a function of the coordinates on this trajectory.

The quantity I = f nds = c At is called the optical path. If the light propagates into isotropic - i.e. homogeneous - successive media, the rays follow straight lines with successive deviations at surface boundaries. Postulating that At is minimum entails that I is the shortest optical path. This is known as the second formulation of Fermat's principle. Let us denote Ik the intersection point at the k-th surface boundary so that the only possible optical ray is A...Ik...B, i.e. the least time ray: the intersection points are such as the variation of the optical path if moving them onto their associated surface boundary must be zero. Following Welford [167] in considering generalized curvilinear coordinates drawn onto the boundary surfaces, and denoting these transversal coordinates p k and qk, the stationarity of the optical path means that for the physically possible ray these coordinates must be such that n' r'x n = n rxn.

If A and B are general points of an optical system, for example including a diffraction grating that provides a spectrum for each diffraction order, then the integral of the optical path calculated from (1.12) is not necessarily a minimum but is stationary. The term stationary implies the notion of returning towards the result of constancy of an integral after zeroing this quantity in a calculus of variations. A detailed treatment on stationarity is given in Born and Wolf [17] including the diffraction theory involving use of electromagnetic equations.

Let us apply Fermat's principle to the simple case of two homogeneous media separated by a plane boundary, k = 1, through which the only possible ray - least time ray - passes at point I1 denoted I (Fig. 1.14). Any ray close to AIB, such as the dashed line ray, would not satisfy Eq. (1.12). Without loss of generality, Snell's law may be derived by setting the incident and refracted rays in the z, y plane of a frame where the x,y plane separates the media of indexes n and n' at its origin. Denoting (0,0,zA), (0,y,0), and (0,yB,zB) the coordinates of points A, I, and B respectively, the optical path is

The stationarity condition for a virtual displacement of point I along y axis is dt dy n yB - y

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