---
title: 'The 4 laws of geometrical optics'
---

* **Fermat's principle** *$\Longrightarrow$ the 4 laws of geometrical optics* :

#### Law of reversibility of the path of light.

Optical path and property of stationarity : concept of orientation not used <br>
*$\Longrightarrow$ stationarity property does not depend on the orientation* of the path.

**$\Longrightarrow$**  the **trajectory** *followed by the light* is **indépendant of the direction of propagation along the trajectory**.

#### Law of the rectilinear light trajectory in homogeneous and isotrope media. 

Euclidian space : *straight line = shortest path between 2 points*

**$\Longrightarrow$**  in an **optically homogeneous and isotrope medium**, the *light travels rectilinearly* : the **light rays are straight lines**.

#### The 2 laws of reflection and refraction
! <details markdown=1>
! <summary>
! IF NECESSARY : reminder if the definitions of the angles and refractive indexes used below. 
! </summary>
! $n_{incid}$ : refractive index of the incident light medium.<br>
! $n_{émerg}$ : refractive index of the emergent light medium (so after crossing the surface).<br>
! $i_{incid}$ : incident ray - normal to the surface at the point of impact angle.<br>
! $i_{émerg}$ : emergent ray - normal to the surface at the point of impact angle.<br>
! </details>

For any incident ray impacting a surface :
* The **surface at the point of impact** is *locally a plane*.
* **Plan of incidence** : plane that *contains the incident ray and normal to the surface at the point of impact*.
* **Refracted and reflected rays** are *in the plane of incidence*, on the *side opposite to the incident ray in relation to the normal* at the surface at the impact point.

*Reflection law* : **$i_{réflec} = i_{incid}$**

*Refraction law (Snell-Descartes)* : for any $i_{incid}$  :
* if $\dfrac{n_{incid}}{n_{émerg}}\cdot\sin(i_{incid})\leqslant1$ then **refraction phenomenon** :<br><br>
**$n_{émerg}\cdot sin(i_{émerg})=n_{incid}\cdot sin(i_{incid})$**<br>

* if $\dfrac{n_{incid}}{n_{emerg}}\cdot\sin(i_{incid})>1$ then **total reflection phenomenon** :<br>
*reflected ray* on the interface that follows the reflection law **$i_{réflec} = i_{incid}$**<br>

* **Critical angle** (of inidence) **for total reflection : $i_{incid_limit}=\arcsin\left (\dfrac{n_{émerg}}{n_{incid}}\right)$** *$\Longrightarrow i_{émerg}=\pi/2\:rad = 90 °$*

_Phenomena of reflection and refraction on a refracting surface._

![](interaction_lumiere_surface_3_650.gif)

!! <details markdown=1>
!! <summary>
!! TO GO FURTHER : intensity distribution between reflected and transmitted beam at a refracting surface.
!! </summary>
!! Geometrical optics: does not quantify the reflected $R$ and transmitted $T$ parts of the  incident beam intensity at a plane refracting surface. This distribution varies according to the incidence angle, the polarization of the incident light, the wavelength. This is described by electromagnetism.
!! However a simple result is useful and to know :
!! *The light intensity is either reflected or transmitted* : $R+T=1$.
!!
!! For a light beam of wavelength $\lambda$ of normal incidence upon a refracting surface :
!! - ratio reflected power versus incident power : $R=\left(\dfrac{n_{incid}-n_{émerg}}{n_{incid}+n_{émerg}}\right)^2$
!! - ratio transmitted power versus incident power  : $T=1-R$
!!</details>

_Total reflection phenomenon_

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![](Opt_Geo_refle_lim_650.gif)




