Fermat's principle

In optics, Fermat's principle, or the principle of least time, named after French mathematician Pierre de Fermat, is the principle that the path taken between two points by a ray of light is the path that can be traversed in the least time. This principle is sometimes taken as the definition of a ray of light. However, this version of the principle is not general; a more modern statement of the principle is that rays of light traverse the path of stationary optical length with respect to variations of the path. In other words, a ray of light follows the path such that there are other paths, arbitrarily nearby on either side, along which the ray would take almost exactly the same time to traverse.the optical length of the path followed by light between two fixed points, A and B, is an extremum. The optical length is defined as the physical length multiplied by the refractive index of the material.'... The principle which you take as the basis for your proof, namely that Nature always acts by using the simplest and shortest paths, is merely a moral, and not a physical one. It is not, and cannot be, the cause of any effect in Nature.Le principe que vous prenez pour fondement de votre démonstration, à savoir que la nature agit toujours par les voies les plus courtes et les plus simples, n’est qu’un principe moral et non point physique, qui n’est point et qui ne peut être la cause d’aucun effet de la nature. In optics, Fermat's principle, or the principle of least time, named after French mathematician Pierre de Fermat, is the principle that the path taken between two points by a ray of light is the path that can be traversed in the least time. This principle is sometimes taken as the definition of a ray of light. However, this version of the principle is not general; a more modern statement of the principle is that rays of light traverse the path of stationary optical length with respect to variations of the path. In other words, a ray of light follows the path such that there are other paths, arbitrarily nearby on either side, along which the ray would take almost exactly the same time to traverse. Fermat's principle can be used to describe the properties of light rays reflected off mirrors, refracted through different media, or undergoing total internal reflection. It follows mathematically from Huygens' principle (at the limit of small wavelength). Fermat's text Analyse des réfractions exploits the technique of adequality to derive Snell's law of refraction and the law of reflection. Fermat's principle has the same form as Hamilton's principle and it is the basis of Hamiltonian optics. The time T a point of the electromagnetic wave needs to cover a path between the points A and B is given by: c is the speed of light in vacuum, ds an infinitesimal displacement along the ray, v = ds/dt the speed of light in a medium and n = c/v the refractive index of that medium, t 0 {displaystyle t_{0}} is the starting time (the wave front is in A), t 1 {displaystyle t_{1}} is the arrival time at B. The optical path length of a ray from a point A to a point B is defined by: and it is related to the travel time by S = cT. The optical path length is a purely geometrical quantity since time is not considered in its calculation. An extremum in the light travel time between two points A and B is equivalent to an extremum of the optical path length between those two points. The historical form proposed by Fermat is incomplete. A complete modern statement of the variational Fermat principle is that .mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 40px}.mw-parser-output .templatequote .templatequotecite{line-height:1.5em;text-align:left;padding-left:1.6em;margin-top:0} In the context of calculus of variations this can be written as In general, the refractive index is a scalar field of position in space, that is, n = n ( x 1 , x 2 , x 3 ) {displaystyle n=nleft(x_{1},x_{2},x_{3} ight)} in 3D euclidean space. Assuming now that light has a component that travels along the x3 axis, the path of a light ray may be parametrized as s = ( x 1 ( x 3 ) , x 2 ( x 3 ) , x 3 ) {displaystyle s=left(x_{1}left(x_{3} ight),x_{2}left(x_{3} ight),x_{3} ight)} and where x ˙ k = d x k / d x 3 {displaystyle {dot {x}}_{k}=dx_{k}/dx_{3}} . The principle of Fermat can now be written as