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| title | media_order | published | routable | visible |
|---|---|---|---|---|
| The Fermat's principle F | stationnarite3_650.jpg | false | false | false |
!!!! IN CONSTRUCTION ! !!!! !!!! very imperfect course, !!!i not validated by the teaching team
Optical path
optical path $\delta$ $=$ euclidean length $s$ $\times$ 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 ${\Longleftrightarrow}:::::\mathrm{d}{\Large\tau}(\Gamma_o)=\sum_i\frac{\partial{\large\tau}}{\partial\lambda_i}(\Gamma_o);\mathrm{d}\lambda_i=0$
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 raysfrom A go through B : here 3 rays :
!!!! Be careful :
!!!! 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 .
- other positions of A and B : 1 maximum $\Longrightarrow$ 1 unique ray from A goes trhough B .
Elliptical concave mirror.
- Elliptical mirror : mirror whose surface is part of an ellipsoid of revolution.
!
! Ellipsoid and ellipsoid of revolution !
! 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. !- between the two geometrical foci" F et F' of an elliptical mirror, all path are stationary : they have the same optical path
$\Longrightarrow$ : *all the rays coming from one of the two foci and intercepting the mirror converge towards the second geometric focus *.
!!!! 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.