Fermat's principle

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Fermat's principle leads to Snell's law; when the sines of the angles in the different media are in the same proportion as the propagation velocities, the time to get from P to Q is minimized.

In optics, Fermat's principle or the principle of least time 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.[1] 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.[2] In other words, a ray of light prefers 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.[3]

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[4] and the law of reflection.

Fermat's principle has the same form as Hamilton's principle and it is the basis of Hamiltonian optics.

Modern version[edit]

The time T a point of the electromagnetic wave needs to cover a path between the points A and B is given by:

T=\int_{\mathbf{t_0}}^{\mathbf{t_1}} \, dt = \frac{1}{c} \int_{\mathbf{t_0}}^{\mathbf{t_1}} \frac{c}{v} \frac{ds}{dt}\, dt = \frac{1}{c} \int_{\mathbf{A}}^{\mathbf{B}} n\, ds\

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 is the starting time (the wave front is in A), 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:

S=\int_{\mathbf{A}}^{\mathbf{B}} n\, ds\

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 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."[5]

In the context of calculus of variations this can be written as

\delta S= \delta\int_{\mathbf{A}}^{\mathbf{B}} n \, ds =0

In general, the refractive index is a scalar field of position in space, that is, n=n\left(x_1,x_2,x_3\right) \ 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=\left(x_1\left(x_3\right),x_2\left(x_3\right),x_3\right) \ and

nds=n \frac{\sqrt{dx_1^2+dx_2^2+dx_3^2}}{dx_3}dx_3=n \sqrt{1+\dot{x}_1^2+\dot{x}_2^2} \ dx_3

where \dot{x}_k=dx_k/dx_3. The principle of Fermat can now be written as

\delta S= \delta\int_{x_{3A}}^{x_{3B}} n\left(x_1,x_2,x_3\right) \sqrt{1+\dot{x}_1^2+\dot{x}_2^2}\, dx_3
= \delta\int_{x_{3A}}^{x_{3B}} L\left(x_1\left(x_3\right),x_2\left(x_3\right),\dot{x}_1\left(x_3\right),\dot{x}_2\left(x_3\right),x_3\right)\, dx_3=0

which has the same form as Hamilton's principle but in which x3 takes the role of time in classical mechanics. Function L\left(x_1,x_2,\dot{x}_1,\dot{x}_2,x_3\right) is the optical Lagrangian from which the Lagrangian and Hamiltonian (as in Hamiltonian mechanics) formulations of geometrical optics may be derived.[6]

Derivation[edit]

Classically, Fermat's principle can be considered as a mathematical consequence of Huygens' principle. Indeed, of all secondary waves (along all possible paths) the waves with the extrema (stationary) paths contribute most due to constructive interference. Suppose that light waves propagate from A to B by all possible routes ABj, unrestricted initially by rules of geometrical or physical optics. The various optical paths ABj will vary by amounts greatly in excess of one wavelength, and so the waves arriving at B will have a large range of phases and will tend to interfere destructively. But if there is a shortest route AB0, and the optical path varies smoothly through it, then a considerable number of neighboring routes close to AB0 will have optical paths differing from AB0 by second-order amounts only and will therefore interfere constructively. Waves along and close to this shortest route will thus dominate and AB0 will be the route along which the light is seen to travel.[7]

Fermat's principle is the main principle of quantum electrodynamics where it states that any particle (e.g. a photon or an electron) propagates over all available (unobstructed) paths and the interference (sum, or superposition) of its wavefunction over all those paths (at the point of observer or detector) gives the correct probability of detection of this particle (at this point). Thus the extremal (shortest, longest or stationary) paths contribute into this interference most as they can not be completely canceled out.

In the classic mechanics of waves, Fermat's principle follows from the extremum principle of mechanics (see variational principle).

History[edit]

Hero of Alexandria (Heron) (c. 60) described a principle of reflection, which stated that a ray of light that goes from point A to point B, suffering any number of reflections on flat mirrors, in the same medium, has a smaller path length than any nearby path.[8]

Ibn al-Haytham (Alhacen), in his Book of Optics (1021), expanded the principle to both reflection and refraction, and expressed an early version of the principle of least time. His experiments were based on earlier works on refraction carried out by the Greek scientist Ptolemy[9]

Pierre de Fermat

The generalized principle of least time in its modern form was stated by Fermat in a letter dated January 1, 1662, to Cureau de la Chambre.[10] It was met with objections made in May 1662 by Claude Clerselier, an expert in optics and leading spokesman for the Cartesians at that time. Amongst his objections, Clerselier states:

... Fermat's principle can not be the cause, for otherwise we would be attributing knowledge to nature: and here, by nature, we understand only that order and lawfulness in the world, such as it is, which acts without foreknowledge, without choice, but by a necessary determination.

The original French, from Mahoney, is as follows:

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.

Indeed Fermat's principle does not hold standing alone, we now know it can be derived from earlier principles such as Huygens' principle.
Historically, Fermat's principle has served as a guiding principle in the formulation of physical laws with the use of variational calculus (see Principle of least action).

See also[edit]

Notes[edit]

  1. ^ Arthur Schuster, An Introduction to the Theory of Optics, London: Edward Arnold, 1904 online.
  2. ^ Ghatak, Ajoy (2009), Optics (4th ed.), ISBN 0-07-338048-2 
  3. ^ Feynman, Richard, The Feynman Lectures on Physics, Vol. 1, pp. 26–7 
  4. ^ Katz, Mikhail G.; Schaps, David; Shnider, Steve (2013), Almost Equal: The Method of Adequality from Diophantus to Fermat and Beyond, Perspectives on Science 21 (3): 7750, arXiv:1210.7750, Bibcode:2012arXiv1210.7750K 
  5. ^ R. Marques, F. Martin, and M. Sorolla. Metamaterials with Negative Parameters. Wiley, 2008.
  6. ^ Julio Chaves, Introduction to Nonimaging Optics, CRC Press, 2008 (ISBN 978-1420054293)
  7. ^ Ariel Lipson, Stephen G. Lipson, Henry Lipson, Optical Physics 4th Edition, Cambridge University Press, ISBN 978-0-521-49345-1.
  8. ^ History of Geometric Optics/Richard Fitzpatrick
  9. ^ Pavlos Mihas (2005). Use of History in Developing ideas of refraction, lenses and rainbow, Demokritus University, Thrace, Greece.
  10. ^ Michael Sean Mahoney, The Mathematical Career of Pierre de Fermat, 1601-1665, 2nd edition (Princeton University Press, 1994), p. 401