In this course, we will only talk about homogenous medium for sake of simplicity.
Definition
Definition of an anisotropic medium
An anisotropic medium is a medium in which the refractive index varies depending on the direction of light propagation.
We can mathematicaly translate this behavior by consider the medium dielectric permittivity (Permittivité d'un milieu) \(\epsilon_r\) as a tensor \(\bar{\bar \epsilon_r}\) (Tenseur).
Permittivité d'un milieu (Milieu anisotrope)
Types of anisotropic medium
To define the types of medium, let's check the \(3\) refraction index defined by the egenvalues of the dieletric permittivity tensor \(\bar{\bar \epsilon_r}\) :- \(n_1\neq n_2\neq n_3\) : the medium is called biaxial.
- \(n_1=n_2\neq n_3\) : the medium is called uniaxial. There is an optical axis defined by the principal direction of \(n_3\).
- \(n_1=n_2=n_3\) : the medium is isotropic.
Ordinary and extraordinary refraction index
In a uniaxial anistropic medium (\(n_1=n_2\neq n_3\)) we can define two refraction index.
We call \(n_3\) the extraordinary index and \(n_1=n_2=n_0\) the ordinary index.
Plane wave in anistropic medium
Vibration and wave plane
Let's define a plane wave: \(\vec E=\vec E_0e^{i(\vec k?\vec r(\omega t))}\)
Equations de Maxwell gives us the following relations:
- \(i\vec k\wedge \vec E=i\omega\vec B\) and \(\vec k.\vec D=0\)
- \(i\vec k\wedge\vec B=-i\mu_0\omega\vec D\) and \(\vec k.\vec B=0\)
Therefore, we can define two plane:- \((\vec B,\vec D)\) is the wave plane where \(\vec k\perp (\vec B,\vec D)\)
- \((\vec k,\vec D)\) is the vibration plane where \(\vec E\perp(\vec k,\vec D)\)
The ray direction is given by the direction of the Poynting vector (Vecteur de Poynting).
Fresnel equation
The Fresnel equation gives the phase velocity a plane wave can have during is propagation in the direction \(\vec u=\frac{\vec k}{||\vec k||}\) in a medium:
$$\frac{u_1^2}{v_\phi^2-v_1^2}+\frac{u_2^2}{v_\phi^2-v_2^2}+\frac{u_3^2}{v_\phi^2-v_3^2}=0$$
With:- \(u_i\) : the principal directions
- \(v_i\) : the principal velocity \(v_i=\frac{c}{n_i}\)
- \(v_\phi\) : the phase velocity of the plane wave \(v_\phi=\frac{\omega}{||\vec k||}\)
Thus, for all directions of propagation \(\vec{u}\), the phase velocity can only assume two values. These directions gives gives directions in which propagation occurs as in an isotropic medium.
This also means that only two directions of vibration are possible. In other words, every plane wave will be decompose on these two directions of polarization (linear).
Birefringence
Birefringence occurs in anisotropic materials when the refractive index depends on the polarization direction of light. This causes an incident light wave to split into two waves, each propagating at different speeds and with different refractive indices. This results in a phase difference between the two waves, altering the polarization of the transmitted light.
Behavior of Different Polarizations
- For a randomly polarized wave: The incident wave is decomposed into two components with different polarizations. Each component interacts with the material in a way that leads to two separate waves, each traveling at different speeds (one experiencing the ordinary index and the other the extraordinary index). This causes birefringence and a phase shift between the waves.
- For a wave polarized along a principal direction: If the wave is polarized along one of the material's principal axes (e.g., the optic axis), the refractive index is unique for that polarization direction. Therefore, the wave does not split into two components, and there is no birefringence. The wave simply propagates according to the specific refractive index for that polarization direction.
Constructions
Huygens method
Https://www.youtube.com/watch?v=UyAO4-V3VfM
With:- purpel lines : light rays
- red curve : extraordinary velocity surface
- blue curve : ordinary velocity surface
- green curve : incidental velocity surface
A velocity surface is define as \(v=\frac c {n(\theta)}\) and \(\theta\) the angular position.
Descartes method
Https://www.youtube.com/watch?v=K-SMd5CGYNI
With:- purpel lines : light rays
- red curve : extraordinary index surface
- blue curve : ordinary index surface
- green curve : incidental index surface
Source : Préparation a l’Agrégation de Sciences Physiques (ENS Cachan), Cours d’optique anisotrope (version 2.0) F. Treussart ; janvier 2003
