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) \cdot A \cdot 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) = (\begin{matrix} cos α & - sin α \\ sin α & cos α \end{matrix}) \cdot (\begin{matrix} e^{- i π / 2} & 0 \\ 0 & e^{i π / 2} \end{matrix}) \cdot (\begin{matrix} cos α & sin α \\ - sin α & cos α \end{matrix})

This utilises the fact that $cos (- α) = cos α$ , $sin (- α) = - sin α$ .

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

\begin{array}{l} A (90 °) = (\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}) \cdot (\begin{matrix} e^{- i π / 2} & 0 \\ 0 & e^{i π / 2} \end{matrix}) \cdot (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}) \\ = (\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}) \cdot (\begin{matrix} - i & 0 \\ 0 & i \end{matrix}) \cdot (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}) \\ = (\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}) \cdot (\begin{matrix} 0 & - i \\ - i & 0 \end{matrix}) \\ = (\begin{matrix} i & 0 \\ 0 & - i \end{matrix}) \end{array}

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

(\begin{matrix} i & 0 \\ 0 & - i \end{matrix}) \cdot (\begin{matrix} 1 \\ 0 \end{matrix}) = (\begin{matrix} i \\ 0 \end{matrix}) = e^{i π / 2} (\begin{matrix} 1 \\ 0 \end{matrix})

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

(\begin{matrix} i & 0 \\ 0 & - i \end{matrix}) \cdot \frac{1}{\sqrt{2}} (\begin{matrix} 1 \\ 1 \end{matrix}) = \frac{1}{\sqrt{2}} (\begin{matrix} i \\ - i \end{matrix}) = \frac{1}{\sqrt{2}} e^{i π / 2} (\begin{matrix} 1 \\ - 1 \end{matrix})

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

c) linear along the z axis

(\begin{matrix} i & 0 \\ 0 & - i \end{matrix}) \cdot (\begin{matrix} 0 \\ 1 \end{matrix}) = (\begin{matrix} 0 \\ - i \end{matrix}) = e^{- i π / 2} (\begin{matrix} 0 \\ 1 \end{matrix})

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

(\begin{matrix} i & 0 \\ 0 & - i \end{matrix}) \cdot \frac{1}{\sqrt{2}} (\begin{matrix} 1 \\ - 1 \end{matrix}) = \frac{1}{\sqrt{2}} (\begin{matrix} i \\ i \end{matrix}) = \frac{1}{\sqrt{2}} e^{i π / 2} (\begin{matrix} 1 \\ 1 \end{matrix})

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 $λ / 4$ retarder can be investigated. He is used for preparing circular polarisation from linear one, and the other way around.

Understanding Spectra from the Earth

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.

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:

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.

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: