Principle of Fermat

The principle of Fermat is a physical principle which is used as base with the geometrical Optique. It describes the form of the optical Chemin of a luminous ray and is stated as follows:

It makes it possible to find the majority of the results of geometrical optics, in particular the laws of the reflection on the mirrors, the laws of the refraction, etc

A consequence first of the principle of Fermat is the rectilinear propagation of the luminous rays in the homogeneous mediums. Indeed, in a homogeneous medium, run time is proportional to the length of the way, and the shortest way to go from a point to another is and was always the straight line.

History

This principle owes its name with Pierre de Fermat, which stated it in 1657 but which did not subject its report, Synthèse for the refractions that in 1662. He is opposed, by that, with Rene Descartes, which in its dioptric , explained the laws of optics by comparing the light with a ball subjected to various forces. Fermat is based on a moral principle: “Nature always acts by the shortest ways and simplest”

Pierre Louis Moreau de Maupertuis into complete the design, creating the Principle of less action .

Preliminary discussion

The principle of Fermat was stated above in its common form but one must state it in the more rigorous form:

Indeed if, in the majority of the cases, one meets ways of which the duration of course is minimal, it should not be forgotten in so far as other situations can exist. For example, to go from a point has to a point B located inside the concave mirror represented opposite, while reflecting itself only once on surface, the light can borrow two courses of minimum duration (in red) and a course of maximum duration (in green). The study length of the way $AMB\; =\; AM\; +\; MB$ would reveal three extrema local, two minima and a maximum, corresponding to these ways.

The stationary term can be included/understood starting from this example. If the unspecified point M carries out first order an infinitely small displacement on the surface of the mirror, the variation of the optical way is also first order. On the other hand, if one considers the three zones which correspond to the extrema, then a first order displacement of the point still involves a variation of the optical of the second order or weaker way. In other words, when the point M moves on the mirror, the variation of the optical way is fast almost everywhere but very slow in the vicinity of the three particular points where it becomes quasi null, from where use of the “stationary” word.

The same problem finds in the statement of the less principle of ''' ''' action, which is with mechanics what is the principle of Fermat to optics, in the definition of the minimal surfaces ''' ''' (which are not inevitably… minimal) or in that of the Géodésique S.

Formulation of the principle of Fermat using the optical way

Case of a homogeneous medium

The optical way of the light to go from a point $A$ towards a point $B$ in a homogeneous medium, noted $L\; (has,\; B)$, is defined as being a number proportional to the time put by the ray to go from $A$ to $B$, the proportionality factor being such as $L\; (has,\; B)$ is equal to the distance $AB$ for a course in the vacuum. Calling $v$ the celerity of the light in the medium, and $c$ that in the vacuum, one thus has $L\; (AB)\; =\; \backslash \; alpha\; \backslash \; tfrac\; \{AB\}\; \{v\}$ and $\backslash \; alpha\; \backslash \; tfrac\; \{AB\}\; \{C\}\; =AB$, from where $L\; (has,\; B)=\; \backslash \; tfrac\; \{C\}\; \{v\}\; AB$; the proportionality factor $n=\; \backslash \; tfrac\; \{C\}\; \{v\}$ is called the Index of refraction medium. The optical way is thus defined by:

$L\; (has,\; B)\; =\; N\; \backslash \; cdot\; AB$ The light is propagated “with difficulty” in the mediums other than the vacuum. Example: A luminous ray crosses a layer of 5 cm thickness water. In parallel, another luminous ray crosses 5 cm of air. Water with for index of refraction $n=1,33$ and air an index appreciably equal to that of the vacuum $n=1$. In water, the luminous ray will have traversed a distance $D=1,33.5=6,65\; \backslash \; mathrm\; \{cm\}$. In the air, the ray will have traversed a distance $D\text{'}=1.5=5\; \backslash \; mathrm\; \{cm\}$. The ray will have traversed a longer optical way in water than in the air.

Case of an unspecified continuous medium

One considers two points infinitely close and distant of a distance $\backslash \; mathrm\; ds$. The optical way separating these two points is defined by $\backslash \; mathrm\; dL=n.\; \backslash \; mathrm\; ds$; $\backslash \; mathrm\; dL$ is the optical differential of way or, the infinitesimal unit element of optical way. To find the way optical $L\; (AB)$ separating two points $A$ and $B$ on this curve, it is enough to make the integral sum of all the elements $\backslash \; mathrm\; dL$ on the curvilinear coordinate $s$ delimited by the points $A$ and $B$:

$L\; (AB)\; =\; \backslash \; int\_\; \{AB\}\; N\; \backslash ,\; \backslash \; mathrm\; ds$

The principle of Fermat is stated then:

Consequences

Principle of the return reverses light

The stationnarity of the optical way makes it possible to proceed to some considerations. Between two points has and B, both on the curvilinear trajectory of a luminous ray being propagated in an inhomogenous medium, one can express the optical integral of way as it follows:

$L\; (AB)\; =\; \backslash \; int\_\; \{AB\}\; N\; \backslash ,\; \backslash \; mathrm\; ds=\; \backslash \; int\_\; \{AB\}\; N\; \backslash ,\; (-\; \backslash \; mathrm\; ds)\; =\; \backslash \; int\_\; \{AB\}\; N\; \backslash ,\; \backslash \; mathrm\; ds\text{'}$

By considering that $\backslash \; mathrm\; ds\text{'}\; =\; -\; \backslash \; mathrm\; ds$ is the curvilinear element of coordinate of B towards has, one can then write $L\; (AB)\; =L\; (BA)$. In the case of the study of an optical system, that means that one will be able to study the light propagation and the way of the luminous rays without worrying about the direction of propagation.

Caution: in practice, the unwise use of this principle leads to nonsenses. It is known for example that an objective never gives a specific image of a luminous point, but a spot always more or less fuzzy. It would be childish to think that the rays forming this spot, returned in the objective by a plane mirror, could reform other side an point-image Net… It is not the principle which is in question here, but the fact of wanting to apply it in a faulty way.

Laws of Snell-Descartes

A $P$ plan of normal vector $\backslash \; vec\; n$ separates two mediums from indexes of refractions $n\_1$ and $n\_2$. A luminous ray starts from a point $A\_1$ located in the first medium, crosses $P$ in $M$ and reaches a point $A\_2$. The problem is to find the position of $M$ starting from the principle of Fermat.

The optical way of $A\_1$ with $A\_2$ is $L=\; n\_1A\_1M+n\_2A\_2M$ thus

.

The condition of stationnarity of $L$, $\backslash \; mathrm\; dL=0$, thus results in the fact that $n\_1\; \backslash \; tfrac\; \{\backslash \; overrightarrow\; \{A\_1M\}\}\; \{A\_1M\}\; +n\_2\; \backslash \; tfrac\; \{\backslash \; overrightarrow\; \{A\_2M\}\}\; \{A\_2M\}$ is orthogonal in the $P$ plan, that is to say colinéaire with $\backslash \; vec\; n$; the luminous ray is thus included in the plan passing by $A\_1$ and $A\_2$ and orthogonal with $P$ (first law of Descartes) and $\backslash \; left\; (n\_1\; \backslash \; tfrac\; \{\backslash \; overrightarrow\; \{A\_1M\}\}\; \{A\_1M\}\; +n\_2\; \backslash \; tfrac\; \{\backslash \; overrightarrow\; \{A\_2M\}\}\; \{A\_2M\}\; \backslash \; right)\; \backslash \; wedge\; \backslash \; vec\; N\; =\; \backslash \; vec\; 0$ gives: $n\_1\; \backslash \; sin\; i\_1\; =n\_2\; \backslash \; sin\; i\_2$ (second law of Descartes).

This demonstration is due primarily to Maupertuis in 1744, nearly one century after the statement by Fermat of its principle.

A pretty illustration is given by it by the problem known as “of the swimming instructor”. This one, located on the beach in $A\_1$ must go to help one drowned located in $A\_2$. Like it runs more quickly than it does not swim, its optimal way (in time) is that which follows the laws of the refraction, that us has just determined.

It is shown that if the surface of separation of the two mediums is an unspecified surface, the two laws of Descartes are preserved (but there can be several possible ways).

Generalization of the laws of Descartes to the case of an unspecified continuous medium

Case of a plane way

By bringing back the plan to a reference mark $Oxy$, one has $L\; (AB)\; =\; \backslash \; int\_\; \{AB\}\; N\; \backslash \; mathrm\; ds=\; \backslash \; int\_\; \{has\}\; ^\; \{B\}\; N\; \backslash \; sqrt\; \{1\; +\; y\text{'}^2\}\; \backslash \; mathrm\; dx=\; \backslash \; int\_\; \{has\}\; ^\; \{B\}\; F\; (X,\; there,\; y\text{'})\; \backslash \; mathrm\; dx$; the equation of Euler-Lagrange of the Calcul of the variations expressing the stationnarity of this integral is written $\backslash \; tfrac\; \{\backslash \; mathrm\; D\}\; \{\backslash \; mathrm\; dx\}\; \backslash \; left\; (\backslash \; tfrac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; y\text{'}\}\; \backslash \; right)\; =\; \backslash \; tfrac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; there\}$. We will solve the particular case where $n$ depends only on $x$: one obtains then $\backslash \; tfrac\; \{\backslash \; mathrm\; D\}\; \{\backslash \; mathrm\; dx\}\; \backslash \; Bigl\; (\backslash \; tfrac\; \{ny\text{'}\}\; \{\backslash \; sqrt\; \{1+\; y\text{'}^2\}\}\; \backslash \; Bigr)\; =0$, in other words $N\; \backslash \; tfrac\; \{\backslash \; mathrm\; Dy\}\; \{\backslash \; mathrm\; ds\}\; =\; \backslash \; rm\; cte$; however if one indicates by $i$ the angle which the tangent with the curve with $Ox$ forms, this condition is written: $N\; \backslash \; cdot\; \backslash \; sin\; i=\; \backslash \; rm\; cte$

This constitutes to some extent an infinitesimal version of the second law of Descartes, which is in fact a local translation of the principle of Fermat.

The differential equation of the trajectory is thus: $y\text{'}=\; \backslash \; tfrac\; \{K\}\; \{\backslash \; sqrt\; \{n^2-k^2\}\}$.

One finds obviously that for $n$ constant, the trajectories are lines; but if one takes $n\; =\; \backslash \; left\; (\backslash \; tfrac\; \{has\}\; \{X\}\; \backslash \; right)\; ^\; \{1\; \backslash \; alpha\}$, the curve paramétrise simply in

$\backslash \; begin\; \{boxes\}\; x=\; \{has\; \backslash \; over\; k^\; \backslash \; alpha\}\; (\backslash \; sin\; T)\; ^\; \backslash \; alpha\; \backslash \; \backslash \; y=\; \{has\; \backslash \; alpha\; \backslash \; over\; k^\; \backslash \; alpha\}\; \backslash \; int\; (\backslash \; sin\; T)\; ^\; \backslash \; alpha\; \backslash \; mathrm\; dt\; \backslash \; end\; \{boxes\}$.

It is a curve of Ribaucour.

For example for $\backslash \; alpha=1$ (the index of refraction is inversely proportional to the X-coordinate) one obtains

$\backslash \; begin\; \{boxes\}\; x=\; \{has\; \backslash \; over\; K\}\; \backslash \; sin\; T\; \backslash \; \backslash \; there\; =\; \{has\; \backslash \; over\; K\}\; \backslash \; cos\; T\; +\; B\; \backslash \; end\; \{boxes\}$, in other words the trajectories are circles.

For $\backslash \; alpha=2$ (the index of refraction is inversely proportional to the square root of the X-coordinate) one obtains

$\backslash \; begin\; \{boxes\}\; X\; =\; \{has\; \backslash \; over\; 2k\}\; (1\; \backslash \; cos\; 2\; T)\; \backslash \; \backslash \; there\; =\; \{has\; \backslash \; over\; 2k\}\; (2t\; -\; \backslash \; sin\; 2t)\; +\; B\; \backslash \; end\; \{boxes\}$: the curve is a Cycloïde. This case is historically interesting: it is by this method which Jean Bernoulli proved that the curve Brachistochrone (minimizing the run time of a material point) is the cycloid one, by showing that the problem of mechanics and that of optics are in fact equivalents.

General case

One now considers a continuous inhomogenous medium with 3 dimensions. It is shown that the generalized laws of Descartes, resulting from the principle of Fermat, are stated in the following form:

1. the osculatory plan with the trajectory in a point $M$ contains the normal on the surface of constant index of refraction passing by $M$.

2. If one indicates by $i$ the angle which the tangent with the curve with this normal makes, one with the relation: $\backslash |\backslash \; overrightarrow\; \backslash \; operatorname\; \{grad\}\; (N)\; \backslash |\; \backslash \; sin\; I\; =\; \backslash \; tfrac\; \{N\}\; \{R\}$ where $R$ is the radius of curvature of the curve.

Random links:

(2030) Belyaev | Maser (Italy) | Emskiant | Saint-Joseph (parish of Tholy) | Projective limit | OpenGL++

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org