Supplement 4.4: Polarisation of Electromagnetic Waves: Jones Vectors and Jones Matrices
Solutions to the task on page 2

1. Please calculate the Jones Matrix of a λ/2 retarder with the fast axis in the direction of the z coordinate.

The equation of a component which is exposed to a light beam at an angle of α is:

A(α)=R(α)⋅A⋅R(−α)

Inserting the matrix of the λ/2 retarder with the fast axis in the direction of the y coordinate, and the rotation matrix:

A(α)=( cosα −sinα sinα cosα )⋅( e −iπ/2 0 0 e iπ/2 )⋅( cosα sinα −sinα cosα )

This utilises the fact that cos(−α)=cosα , sin(−α)=−sinα .

With α=90° for the orientation of the fast axis in the direction of the z coordinate it follows:

A(90°)=( 0 −1 1 0 )⋅( e −iπ/2 0 0 e iπ/2 )⋅( 0 1 −1 0 )                =( 0 −1 1 0 )⋅( −i 0 0 i )⋅( 0 1 −1 0 )                =( 0 −1 1 0 )⋅( 0 −i −i 0 )                =( i 0 0 −i )

This is the required result.

2. Calculate the types of polarisation of the transmitted light for a λ/2 retarder with the fast axis in direction of the z coordinate, for incident light having the intensity 1 and the following polarisations:

a) linear along the y axis

( i 0 0 −i )⋅( 1 0 )=( i 0 )= e iπ/2 ( 1 0 )

The transmitted light is still linear polarised along the y axis. The phase is delayed by π/2, since the light is passing the retarder with polarisation along the slow axis.

b) linear diagonally in the first and third quadrant

( i 0 0 −i )⋅ 1 2 ( 1 1 )= 1 2 ( i −i )= 1 2 e iπ/2 ( 1 −1 )

The light is linear polarised diagonally along the second and fourth quadrant, the polarisaton is rotated by 90°.

c) linear along the z axis

( i 0 0 −i )⋅( 0 1 )=( 0 −i )= e −iπ/2 ( 0 1 )

The transmitted light is linear polarised along z. The phase is leading by π/2 because of the transmission along the fast axis.

d) linear diagonally in the second and fourth quadrant

( i 0 0 −i )⋅ 1 2 ( 1 −1 )= 1 2 ( i i )= 1 2 e iπ/2 ( 1 1 )

The light is linear polarised diagonally along the first and third quadrant, the polarisaton is rotated by 90°.

Throughout these results, the λ/2 retarder having the appropriate orientation is preferably used to rotate the orientation of linear polarised light by 90°.

Further questions:
- how does the polarisation of the transmitted light change for other kinds of polarisation than the linear we dealt with?
- how does the polarisation change if the retarder is hit by the beams at a different angle α?
This can be determined by a similar calculation.

In the same way the effect of a λ/2 retarder can be investigated. He is used for preparing circular polarisation from linear one, and the other way around.