Fundamental principles

Linear polarization's state is characterized by an oscillation of the electric field vector which runs on a straight line. Should the wave travel in the z-direction, the straight line drawn by belongs to the xOy plane . The components od Ex and Ey have a phase difference of Ifd m is nil or an even whole number, the components are said to be in phase. If m is an odd whole number, the polarization line is orthogonally oriented compared to the previous one (cf Figure 16).

Circular polarization's state is characterized by its two components Ex and Ey having an identical amplitude and a phase difference as:

Under these circumstances, the extremity of the electric field vector traces a circle either clockwise or anticlockwise-wise. You can determine the way the rotation moves by watching the wave coming. A right-hand circular polarization (i.e. when rotates clockwise) is obtained when

where

By analogy, a phase difference of

where

will give a left-hand circular polarization, will rotate in the anticlockwise direction.

A circular polarization can be only specified by its amplitude and by which way (left or right) rotates.

In any other circumstances, the extremity of the electric field clockwise or anticlockwise traces an ellipse in the xOy plane when rotating (cf Figure 9). The amplitude of the components Ex and Ey is not identical and their phase difference has not characteristic value. Linear and circular polarizations are specific cases of elliptical polarization.

There are several formal representations of the polarization and of the modeling of the transmission of light which has been polarized through a polarizing medium, like birefringent mediums. The most common method is Jones' method, that we will tell in detail later.

Jones' matrices formalism gives us information about light polarization which travels in a complex medium. The state of polarization can be estimated by using matrix algebra [40]. The state of polarization of a wave is represented by two components which are complex numbers as:

where Ex and Ey are the amplitudes and   and  are the phases of components of in the xOy plane.

The previous equation describes the general case of an elliptically polarized wave travelling in z -direction. In the case of a linear polarization which makes an angle with the x - axis , the Jones matrix becomes:

A circular polarization will be described as:

If you take the real numbers out of each component of the wave you find:

When t=0 , the electric field vector, is given by Ex=E and Ey=0, so ift increases, Ex decreases but remains positive and Ey increases but remains negative. Consequently clockwise traces a circle.

Now let's study the matrices which describe the diverse mediums through which light passes. The simplest example is a Jones matrix in an isotropic absorbing medium, which weakens the transmission but does not modify the state of polarization. Such a medium is represented by the matrix:

where  is the weakening of the medium.

An ideal polarizer will only let one direction of the electric field go out and will stop the others. A polarizer oriented according to the x - axis will have as a matrix:

and a wave polarized in an unparticular way will have no more component on the y-axis after having passed through the polarizer:

Generally, the matrix of a polarizer whose transmitting axis has a angle with the x -axis in thexOy plan can be written as:

The state of polarization when coming out of a system made of several elements is obtained by multiplying the state of polarization when entering the system by the series of individual matrices of the diverse elements encountered all along the optical path. The matrix multiplication is not commutative, so be careful with the matrices' order.

An optical retardant is a component made of a birefringent material and is used to change the state of polarization of any incident wave. When entering the device, the wave be divided according to the main axes of the medium, which are called ordinary axis and extraordinary axis. The two components will propagate with different speeds characterized by the ordinary refractive index no and the extraordinary refractive index ne. When coming out of the device, the components have not the same phase differences anymore and so they have not the same states of polarization as before entering anymore too.

The relative phase difference   between the ordinary and extraordinary axes is given by the equation:

where d is the thickness of the material and   is the wavelength.

The thickness of the birefringent material d is chosen in order to produce the expected phase difference. The most famous optical retarders are the half-wave plates and the quarter-wave plates. A half-wave plate makes the direction of a linear polarization rotate by 90°. A quarter-wave plate makes a   phase difference between the components of the light travelling on the ordinary and extraordinary axes.

A quarter-wave plate makes a circular polarization out of a linear polarization which was 45°-oriented at its main axes. Otherwise, a circular polarization will give a linear polarization which is 45°-oriented at the main axes of the plate.