---
title: 'The Fermat''s principle F'
media_order: stationnarite3_650.jpg
---

#### Optical path

**optical path** *$\delta$* &nbsp;&nbsp;&nbsp;&nbsp; $=$
**euclidean length** *$s$* &nbsp;&nbsp;&nbsp; $\times$ &nbsp;&nbsp; **refractive index** *$n$*

* **$\Gamma$** : *path (solid line) between 2 fixed points A and B*
* **$\mathrm{d}s_P$** : *element of infinitesimal length at point P on path $\Gamma$*
* **$ n_P$** : *refractive index at point P*
* **$\mathrm{d}\delta_P$** : *infinitesimal optical path at point P on path $\Gamma$*

Optical path along a path between 2 fixed points A and B :
**$\delta\;=\;\int_{P \in \Gamma}\mathrm{d}\delta_P\;=\;\int_{P \in \Gamma}n_P\cdot\mathrm{d}s_P$**

* **$\delta$** $=\int_{\Gamma}n\cdot\mathrm{d}s\;=\;\int_{\Gamma}\frac{c}{v}\cdot\mathrm{d}s$ = $c\;\int_{\Gamma}\frac{\mathrm{d}s}{v}$ = *$\;c\;\tau$*
* **$\delta$** is *proportional to the travel time*.


#### Stationarity

* **$\Gamma_o$** : *path between two fixed point A and B*
* **$\lambda_i$** : *parameter that defines a path*
* **${\Large\tau}$** : *grandeur physique caractérisant un chemin*
 
**${\Large\tau}(\Gamma_o)$ stationnaire &nbsp;&nbsp;
${\Longleftrightarrow}\:\:\:\:\:\mathrm{d}{\Large\tau}(\Gamma_o)=\sum_i\frac{\partial{\large\tau}}{\partial\lambda_i}(\Gamma_o)\;\mathrm{d}\lambda_i=0$**

![](stationnarite3_400.jpg)

#### Fermat's principle

**Between two points** of its path, a **ray of light** follows **the path(s)** that present(s) a *stationary travelling time*.

or (equivalent )

**Between two points** of its path, a **ray of light** follows **the path(s)** that present(s) a *stationary optical path*.


#### Examples 

##### Spherical concave mirror

* **A** : *point source* that emits or diffuses lights in all directions.
* **B** : *fixed point in space*.

For this mirror, **according to the positions of points A and B** :

* **several extrema** : here *2 maxima* et *1 minimum* **$\Longrightarrow$ several ligh rays**from A go through B : here *3 rays* :

<!--à finaliser si possible :
1) DIFFICILE : mise au point automatique : largeur 100% de l'écran et bonne hauteur 'difficile cela' de toute application geogebra.
2) TRES DIFFICILE : faire en sorte que en mode connexion, l'image animée correspondante au geogebra s'affiche tant que l'animation iframe geogebra n'est pas téléchargée, puis elle ne s'affiche pas lorsque le geogeba est prêt, et que ce soit elle qui s'affiche en mode "hors connexion".-->

[Click here for geogebra animation](https://www.geogebra.org/material/iframe/id/syegm6gp)

<!--image gif correspondante-->
![](Fermat_mir_3ray_650.gif)

!!!! *Be careful* :<br>
!!!! Understand this example of application of Fermat's theorem. It says that *the 3 trajectories drawn between A and B satisfy the Fermat's principle* and therefore *are possible trajectories between these two points* to the exclusion of any other trajectory. If point A is a point source that emits light in all directions, then these 3 paths will be followed by light. If a ray goes through A with one of these 3 directions, then the corresponding trajectory will be realized.
!!!! *But* the points *A and B are not conjugate points within th meaning of paraxial optics* : B is not the image point of point source A, and vice versa.
!!! It is the same for all the animations of this chapter "examples".

* other positions of A and B : **1 minimum $\Longrightarrow$ 1 unique ray** from A goes trhough B .

![](Fermat_mir_1ray_min_650.gif)

* other positions of A and B : **1 maximum $\Longrightarrow$ 1 unique ray** from A goes trhough B .

![](Fermat_mir_1ray_max_650.gif)

##### Elliptical concave mirror. 

* Elliptical mirror : mirror whose surface is part of an ellipsoid of revolution.

! <details markdown=1>
! <summary>
! Ellipsoid and ellipsoid of revolution
! </summary>
! Est-il nécessaire de rapeller ici ce que sont les ellipsoïde et ellipsoïde de révolution? rappel en texte? ou 2 liens vers Wikipédia? ou lien vers une autre page m3p2 sur les quadriques en géométrie euclidienne (page encore à créer) ? Si oui, dans une partie Beyond, parler du miroir elliptique concave acoustique, c'est impressionnant quand on le vit.
! </details>

* **between the two geometrical foci"** F et F' of an elliptical mirror, **all path are stationary** : they have the same optical path<br>
**$\Longrightarrow$** : *all the rays coming from one of the two foci and intercepting the mirror converge towards the second geometric focus *.

[Click here for geogebra animation](https://www.geogebra.org/m/aaus5dpr)

![](fermat_mir_elliptique_650.gif)

!!!! *BE CAREFUL* : 
!!! the "geometric foci" of the ellipsoid of revolution, the "geometric surface" in which the surface of the elliptical mirror is inscribed, do not correspond to the optical foci (the two focal points)  of the elliptical mirror as they will be defined in the "optical sense" of the word "focus" in the remainder of this course. 

