Supplement 3.2: Polarisation of Electromagnetic Waves: Jones Vectors and Jones Matrices (1/2)

 

Polarisation may be depicted easier and more clearly if the y and z components of the electric field are described by vector elements. Interaction of light and matter changes for example its intensity and polarisation due to reflection or refraction. These changes can be easily described through a matrix which interacts with the vector and changes its values. This procedure has been developed by R. Clark Jones, thus the field vectors and the matrices are named Jones Vectors and Jones Matrices.

Field components as column matrix: Jones Vectors

For simplicity and clarity, a description of the wave through a complex exponential function instead of a sine function is helpful:

E (x,t)= E o e i( kxωt )
Complex exponential functions

This is a plane wave propagating in x direction, though the following depiction can also be used for waves with a radial propagation and others. We write the electric field as a column vector with the components in the y,z plane. This is the Jones Vector:

E =( E y E z )=( E y,o e i( kxωt ) E z,o e i( kxωt+φ ) )

Often only the amplitudes and phases of the wave components are of interest instead of the spectral properties of the light when examining polarisation. In that case, one may leave out the exponential function which describes the monochromatic wave and write instead:

( E y,o e i φ y E z,o e i φ z )
Equations

One may simplify even more and equate the intensity with 1 if only the type of polarisation is of interest. The intensity though is proportional to the second power of the field strength as it was shown in the Section about electromagnetic waves. The result of Ey2+Ez2 thus shall be 1. This explains the expressions in the following table of Jones Vectors for various polarisations.

Jones Vector Jones Vector for intensity=1 Type of polarisation
( E y,o e i φ y 0 ) ( 1 0 ) linear along the y axis
( 0 E z,o e i φ z ) ( 0 1 ) linear along the z axis
E y,o e i φ y ( 1 1 ) 1 2 ( 1 1 ) linear diagonally in the first and third quadrant of the y,z plane
E y,o e i φ y ( 1 1 ) 1 2 ( 1 1 ) linear diagonally in the second and fourth quadrant of the y,z plane
E y,o e i φ y ( 1 i ) 1 2 ( 1 i ) right circular
E y,o e i φ y ( 1 i ) 1 2 ( 1 i ) left circular
Example: right circular polarisation
This method uses monochromatic waves and is therefore only applicable to completely polarised light.