Supplement 3.3: Polarisation of Electromagnetic Waves: Stokes Vectors and Müller Matrices (1/2)

Stokes Vectors

Besides the method of R. Clark Jones, another matrice operation has been developed. This one does not use the field components of electromagnetic waves but rather their second power. Four square expressions describe the polarisation state, the degree of polarisation and the light intensity. They trace back to George Gabriel Stokes, for what they are named Stokes Parameters. Comparable to the two components Jones Vector, they form the four components Stokes Vector. Interactions of light are hence described by 4×4 matrices, which have been introduced by Hans Müller so that they are called Müller Matrices.

Intensity components as column matrix: Stokes Vectors

Again we presume an electric field vector of an electromagnetic wave, which propagates in x direction:

\vec{E} = (\begin{matrix} E_{y} \\ E_{z} \end{matrix})

The field components shall be complex functions again. Their second powers may be calculated by multiplying the complex components with the conjugated complex components. Stokes defined four square terms, the so called Stokes Parameters,

\begin{array}{l} I_{y} = E_{y} E_{y}^{*} \\ I_{z} = E_{z} E_{z}^{*} \\ U = E_{y} E_{z}^{*} + E_{z} E_{y}^{*} \\ V = i (E_{y} E_{z}^{*} - E_{z} E_{y}^{*}) \end{array}

which form the elements of the Stokes Vector:

\vec{S} = (\begin{matrix} I_{y} \\ I_{z} \\ U \\ V \end{matrix})

Square of complex quantities ↓

Complex quantities are not easy squared. Their second powers are rather calculated as a product of the complex quantity and its conjugated complex quantity. The conjugated complex quantity is marked by the symbol ∗ and results from the complex quantity when the sign of the imaginary unit $i = \sqrt{- 1}$ is changed. If

z = x + i y

is a complex quantity with the real part $x$ and the imaginary part $y$ , the matching conjugated complex quantity is

z^{*} = x - i y

The second power of $z$ becomes:

z z^{*} = x^{2}

as it is easily shown by expanding under consideration of $i^{2} = - 1$ .

The parameter names lead to the presumption, that $I_{y}$ and $I_{z}$ are the intensites of the light which oscillates in y and z direction. The significance of $U$ and $V$ is not clearly evident. Some examples such as their role are indicated in the right column.

An alternative definition ↓

In literature, another definition of the first two Stokes Parameters is often used:

\begin{array}{l} I = E_{y} E_{y}^{*} + E_{z} E_{z}^{*} = I_{y} + I_{z} \\ Q = E_{y} E_{y}^{*} - E_{z} E_{z}^{*} = I_{y} - I_{z} \\ U = E_{y} E_{z}^{*} + E_{z} E_{y}^{*} \\ V = i (E_{y} E_{z}^{*} - E_{z} E_{y}^{*}) \end{array}

According to that, the Müller Matrices are constructed differently. This has to be kept in mind while searching for matrices in literature.

We presume again a plane monochromatic wave without any restrictions of the generality.

E_{y} = E_{y, o} e^{i (k x - ω t)} E_{z} = E_{z, o} e^{i (k x - ω t + φ)}

The following Stokes Parameters will be revealed:

\begin{array}{l} I_{y} = E_{y,0}^{2} \\ I_{z} = E_{z,0}^{2} \\ U = 2 E_{y,0} E_{z,0} cos φ \\ V = - 2 E_{y,0} E_{z,0} sin φ \end{array}

Hereby, the Stokes Vectors (mentioned in the table) for the most important types of polarisation will follow. The intensity of light is normalised to 1. The term $ℓ$ stands for linear polarisation with an index which indicates the angle $α$ against the y axis. $c$ stands for circular polarisation and $r$ marks unpolarised (natural) light. The arrows above the vectors have been omitted for simplicity.

Equations ↓

Stokes Vector	Type of polarisation
$ℓ_{0} = (\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix})$	Linear along the y axis
$ℓ_{90} = (\begin{matrix} 0 \\ 1 \\ 0 \\ 0 \end{matrix})$	Linear along the z axis
$ℓ_{45} = (\begin{matrix} 1 / 2 \\ 1 / 2 \\ 1 \\ 0 \end{matrix})$	Linear diagonally in the first and third quadrant of the y,z plane.
$ℓ_{135} = (\begin{matrix} 1 / 2 \\ 1 / 2 \\ - 1 \\ 0 \end{matrix})$	Linear diagonally in the second and the fourth quadrant of the y,z plane.
$c_{r} = (\begin{matrix} 1 / 2 \\ 1 / 2 \\ 0 \\ 1 \end{matrix})$	right circular
$c_{l} = (\begin{matrix} 1 / 2 \\ 1 / 2 \\ 0 \\ - 1 \end{matrix})$	left circular
$r = (\begin{matrix} 1 / 2 \\ 1 / 2 \\ 0 \\ 0 \end{matrix})$	unpolarised

Obviously the following properties are valid:

The intensity of light is $I_{y} + I_{z}$

\sqrt{{(I_{y} - I_{z})}^{2} + U^{2} + V^{2}} = I_{y} + I_{z} = 1

For unpolarised light:
$I_{y} - I_{z} = U = V = 0$
For the already defined degree of polarisation
$p$ = (Intensity of the polarised part)/(Total intensity)
the following term will follow:
$p = \frac{\sqrt{{(I_{y} - I_{z})}^{2} + U^{2} + V^{2}}}{I_{y} + I_{z}}$

Stokes Vektors use second powers of the electric field strength. Due to

E E^{*} = E_{o} e^{i (k x - ω t + φ)} E_{o} e^{- i (k x - ω t + φ)} = E_{o}^{2}

the phase information and also the information about the spectral properties will be lost through squaring. Read more in the section about the electromagnetic waves in chapter 1.

Stokes Vectors are therefore not suitable for the analysis of coherent effects. In contrast to Jones Vectors also unpolarised light can be described and the degree of polarisation in partially polarised light can be determined.

Understanding Spectra from the Earth

Supplement 3.3: Polarisation of Electromagnetic Waves: Stokes Vectors and Müller Matrices (1/2)

Stokes Vectors

Intensity components as column matrix: Stokes Vectors