Fermat's Principle - Modern Version

Modern Version

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. 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 to a point B is equivalent to an extremum of the optical path length between those two points. The historical form proposed by French mathematician Pierre de Fermat is incomplete. A complete modern statement of the variational Fermat principle is that

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."

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, in 3D euclidean space. Assuming now that light travels along the x₃ axis, the path of a light ray may be parametrized as and

where . The principle of Fermat can now be written as

which has the same form as Hamilton's principle but in which x₃ takes the role of time in classical mechanics. Function is the optical Lagrangian from which the Lagrangian and Hamiltonian (as in Hamiltonian mechanics) formulations of geometrical optics may be derived.