polarization
Light waves propagation in an homogeneous medium
Electomagnetic waves propagating in the vacuum are characterized by their electric field ( E ) and their magnetic induction ( B ) :
.
These formulas lead to :
.
ε0 and μ0 are respectively the vacuum permittivity and the magnetic constant linked by the relation : 
.
The simplest solution of the equations above is a plane and monochromatic wave ( propagating in a single direction at a single wavelength ).
Using complex notations, its electric field vector is :
.
ω is the wave pulsation.
is the wave vector and gives the direction of propagation.
where λ is the wavelength.
is the vector from the coordinate system origin to the observation point.
E0 is the maximum amplitude of the electric field.
The magnetic induction is :
.
and 
are perpendicular to each other.
As , 
and 
are both perpendicular to the direction of propagation.
, and form a direct trihedron.
The electric field and magnetic induction amplitudes satisfy :
.
In a non absorbing homogeneous medium, the Maxwell equations are changed in :
.
εr is the relative permittivity of the medium.
Therefore, in any non absorbing homogeneous medium, waves propagate similarly than in vacuum. The electric field and the magnetic induction of a monochromatic plane wave are :
,
.
n is the refraction index of the medium :
.
Note that the directions of
and are not necessarily constant over time.
The energy propagates in the same direction than the wave front and is characterized by the Poynting vector :
.
The wave intensity I is the average over time of the Poynting vector norm :
.
Polarization states
According to the formulas detailed in the section above, a plane monochromatic wave propagating along the Z axis can be considered as the superposition of two plane waves propagating along Z and wich electric fields 
and 
are perpendicular to each other ( respectively oriented along 
and 
).
Therefore :
.
.
By changing the time origin, the electric field can be written :
.
.
In the case where the wave is "unpolarized", φ has a random value regading time.
Let consider a polarized wave where φ is constant over time.
The amplitudes of and are thus satisfying :
.
Therefore, in the general case, the extremity of the electric field describes an ellipse over time.
The polarization is said elliptic.
It is possible to define the sens of rotation of the electric field by calculating the EX and EY derivates regarding time :
.
At t = 0 :
.
Therefore, rotates in the positive way (the wave is then said positive wave) if φ > 0. It rotates in the negative way (the wave is then said negative wave) if φ is smaller than 0.
In the particular case where EX0 = EY0 and φ = +/-π/2 :
.
describes a circle and the polarization is said circular.
If φ = 0 or φ = π :
.
remains in a constant direction. The polarization is said linear.
Polarization and dielectric interfaces
The Snell descartes Law describes the reflexion and transmission of a light ray at the interface between two homogeneous media. This law applies also for waves.
The incident wave can be split in two waves : one wave with the electic field perpendicular to the incidence plane (defined as TE polarization - see "glossary") and the second one with the electric field in the incident plane (defined as TM polarization - see "glossary").
In this section, the magnetic field H is used instead of the magnetic induction B. They have the same direction and their amplitudes are linked by the relation :
.
In the following formulas :
- index I is related to the parameters of the incident wave,
- index R is related to the parameters of the reflected wave,
- index T is related to the parameters of the refracted ( transmitted ) wave.
Wirh the same logic, nI and nT are the refraction indexes respectively of the first and second media.
For TE polarization, the boundary conditions at the interface are :
.
This brings to :
.
Therefore, the reflexion and transmission ratios in amplitude at the interface are :
.
For TM polarization, the boundary conditions at the interface are :
.
This brings to :
.
Whatever the polarization, if r ant t are respectively the reflexion and transmission ratio in amplitude, the reflexion and transmission ratios ( respectively R and T ) are given by :
.
The reflexion coefficient is never null whatever the incidence angle for TE polarization. It can be zero for TM polarization for an incidence angle said Brewster angle θB satisfying : 
.
Incidence at Brewter angle is currently used to control the polarization. Indeed, the reflected light from an incident wave at Brewter angle is linearly polarized with an electric field perpendicular to the incidence plane (TE polarization). It is also commonly used in gas lasers. Indeed, the gas tube is closed by windows at Brewter angle. Therefore, the wave travels in the laser cavity without Fresnel reflexions at the windows interface for the TM polarization that can then oscillate more efficiently than TE polarization.
Birefringence
Unlike homogeneous (isotropic) materials, optical properties of birefringent (anisotropic) materials depend on the direction of propagation. Because of their effect on light propagation and polarization, anisotropic materials are used in many optical components as polarizers, retarder plates, birefringent filters, beam splitters,....
In homogeneous materials, electromagnetic waves "see" the same refraction index whatever their propagation direction. It is because electric field , electric induction 
and polarization 
are parallel :
.
χ is the electromagnetic susceptibility.
In birefringent materials, because atoms arrangements have their own symetries, the polarization "sensitivity" to an electric field depends on the electric field direction. Therefore, except in particular cases, the polarization and the electric field are not parallel. The first relation above is changed in :
.
is a 3 X 3 tensor.
The electric induction , which is parallel to the electric field in an homogeneous material, can be written :
.
is the "identity" tensor.
More commonly, this relation can be written 
.
is the dielectric tensor.
As a consequence, the Poynting vector (giving the direction of the light ray) remains perpendicular to the electric field and is in general no more parallel to the wave vector (giving the propagation direction of the wave) that remains perpendicular to the electic induction.
Because of the atoms arrangement symetries, the dielectric tensor has less than 9 undependant coefficients. In a particular coordinate system referred to symetry axis of the material (optical axis), it can be diagonalized as follows :
.
nX, nY and nZ are the refraction indexes respectively related to the X, Y and Z axis.
The resolution of Maxwell equations shows that for a given pulsation ω and a given wave direction, there are two solutions giving two possible values for the wave vector and therefore of the refraction index n, each solution having different linear polarization.
Indeed, the two values of the refraction index n are given by the equation :
.
The associated electrield field is :
.
uX, uY, uZ are the coordinates of the unitary vector 
giving the direction of propagation.
This formuma is only valid in the general case for n different from nX, nY and nZ.
There are two main types of birefringent materials : uniaxial and biaxial materials.
The biaxial materials are corresponding to the general case presented above and will not be discussed further.
Uniaxial material
For uniaxial materials, the dielectric tensor is :
.
The crystal axis ( optical axis ) is the Z axis. no is said the ordinary index and ne the extraordinary index.
In the rest of this section, only uniaxial materials are considered.
The wave vector 
satisfies the relation :
.
.
Therefore, for a given direction of propagation , the two solutions for discussed above are corresponding to the intersection of the direction of propagation with a sphere and with an ellipsoid.
When propagating in the XY plane, a wave has two different propagation mode, one related to the ordinary index polarized in the XY plane and the other related to the extraordinary index, polarized along Z.
When propagating along the Z axis, the wave has only one propagation mode corresponding to the ordinary index and the polarization is in the XY plane.
When propagating in any direction making an angle θ with Z, one mode is obviously related to the ordinary index and is polarized in the XY plane, the other is related to the index n( θ ) satisfying the formula :
.
The polarization can be deducted from the formula above giving the electric field regarding the refraction index. As illustrated in the two pictures above, no can be either smaller ( positive uniaxial material ) or larger than ne ( negative uniaxial material ). (Note that the scheme above are wrong because of a scaling issue. Indeed, the curves representing no should be circular)
At the interface with a birefringent crystal, two refraction occures : one for the ordinary polarization direction and the other for the extraordinary polarization direction. An incident plane wave is therefore refracted in two directions ( modes ). These directions can be calculated using Huygens method: for each refracted wave, the wavefront is tangent to the corresponding light velocities surface as shown on the graphics beside. Note that the direction of the wave vector for the extraordinary polarization is perpendicular to the wave front and is in general not parallel to extraordinary ray direction.
Birefringent plate
The birefringence is commonly used to control the polarization. Let consider a plane wave propagating in the XY plane and incident on a birefringent plate. Its electrical field can be split in a component in the XY plane corresponding to the ordinary index and another component along Z corresponding to the extraordinary index :
.
At the plate output, the electrical field is :
.
there is a phase shift 
between the two polarizations and the electric field can be written :
.
If the plate thickness is such that 
(wave plate), the polarization remains unchanged.
If the thickness is such that 
(half-wave plate) and the incident wave has a linear polarization making an angle θ with the Z axis :
.
Therefore the polarization remains linear and its direction is symetrical to the incident one regarding Z or 
.
If the thickness is such a that 
(quarter wave plate) and the incident wave has a linear polarization making an angle θ with the Z axis :
.
The wave is then elliptically polarized.
If θ = 45° :
.
The polarization is then circular.
References
"Optique Fondements et applications" - 2004 - author : José-Philippe Perez.
"Cours doptique physique" - Institut d'optique théorique et appliquée - 1985 - author : Christian Imbert.
"Cours d'optique ondulatoire" - Université Denis Diderot Paris 7 - 2006 - author : G.Rebmann.
"Études graphiques des propriétés optiques des lames minces" - Journal de physique - 1950 - author : D. Malé.
"Electromagnétisme des milieux continus" - Licence de physique, Institut Gallilée, Université de Paris Nord - 2000 / 20001 - author : P. Kuzel.