Supplement 4.5: Polarisation of Electromagnetic Waves: Stokes Vectors and Müller Matrices
Solutions to the task on page 2

1. Please calculate the Müller Matrix of a $λ / 4$ 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:

M (α) = R (α) \cdot M \cdot R (- α)

In this case:

Q_{λ / 4} (90 °) = R (90 °) \cdot Q_{λ / 4} \cdot R (- 90 °)

It is $cos (- α) = cos α$ , $sin (- α) = - sin α$ . With $α = 90 °$ for the orientation with the fast axis in direction of the z coordinate follows:

\begin{array}{l} R (90 °) \cdot Q_{λ / 4} \cdot R (- 90 °) \\ = (\begin{matrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) \cdot (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & - 1 \\ 0 & 0 & 1 & 0 \end{matrix}) \cdot (\begin{matrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) \\ = (\begin{matrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) \cdot (\begin{matrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 \\ 0 & 0 & - 1 & 0 \end{matrix}) \\ = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & - 1 & 0 \end{matrix}) = Q_{λ / 4} (90 °) \end{array}

This is the required result.

2. Calculate the type of polarisation of the transmitted light when the $λ / 4$ retarder has this orientation, with incoming light of the intensity 1 and the following polarisations:

a) linear along y

Q_{λ / 4} (90 °) \cdot ℓ_{0} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & - 1 & 0 \end{matrix}) \cdot (\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}) = (\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}) = ℓ_{0}'

The outcoming light is still polarised linear along y.

b) linear diagonally in the first and third quadrant

Q_{λ / 4} (90 °) \cdot ℓ_{45} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & - 1 & 0 \end{matrix}) \cdot (\begin{matrix} 1 / 2 \\ 1 / 2 \\ 1 \\ 0 \end{matrix}) = (\begin{matrix} 1 / 2 \\ 1 / 2 \\ 0 \\ - 1 \end{matrix}) = c_{l}'

The outcoming light is of a left cicular polarisation.

c) linear along z

Q_{λ / 4} (90 °) \cdot ℓ_{90} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & - 1 & 0 \end{matrix}) \cdot (\begin{matrix} 0 \\ 1 \\ 0 \\ 0 \end{matrix}) = (\begin{matrix} 0 \\ 1 \\ 0 \\ 0 \end{matrix}) = ℓ_{90}'

The outcoming light is still polarised linear along z.

d) linear diagonally in the second and fourth quadrant

Q_{λ / 4} (90 °) \cdot ℓ_{135} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & - 1 & 0 \end{matrix}) \cdot (\begin{matrix} 1 / 2 \\ 1 / 2 \\ - 1 \\ 0 \end{matrix}) = (\begin{matrix} 1 / 2 \\ 1 / 2 \\ 0 \\ 1 \end{matrix}) = c_{r}'

The outcoming light is polarised in a right circular manner.

Throughout these results, the $λ / 4$ retarder may be used to convert linear polarised light into circular polarised and the other way round.

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.

Understanding Spectra from the Earth

Supplement 4.5: Polarisation of Electromagnetic Waves: Stokes Vectors and Müller Matrices Solutions to the task on page 2

1. Please calculate the Müller Matrix of a λ/4 retarder with the fast axis in the direction of the z coordinate.

2. Calculate the type of polarisation of the transmitted light when the λ/4 retarder has this orientation, with incoming light of the intensity 1 and the following polarisations:

Supplement 4.5: Polarisation of Electromagnetic Waves: Stokes Vectors and Müller Matrices
Solutions to the task on page 2

1. Please calculate the Müller Matrix of a $λ / 4$ retarder with the fast axis in the direction of the z coordinate.

2. Calculate the type of polarisation of the transmitted light when the $λ / 4$ retarder has this orientation, with incoming light of the intensity 1 and the following polarisations: