Supplement 4.5: Polarisation of Electromagnetic Waves: Stokes Vectors and Müller Matrices (2/2)

Optical interactions as 4×4 matrix: Müller Matrices

The elements of a Stokes Vector $\vec{S}$ change as they are multiplied by a 4×4 matrix. This matrix indicates interactions of light with an optical element, or another optical effect which affects intensity and polarisation. When marking the Stokes vector after the interaction - just like the Jones Vector in the preceding section - as a dotted quantity, it may be written:

(\begin{matrix} I_{y}' \\ I_{z}' \\ U' \\ V' \end{matrix}) = (\begin{matrix} a_{11} & ... & ... & a_{14} \\ ... & ... & ... & ... \\ ... & ... & ... & ... \\ a_{41} & ... & ... & a_{44} \end{matrix}) \cdot (\begin{matrix} I_{y} \\ I_{z} \\ U \\ V \end{matrix}) \Leftrightarrow \vec{S}' = M \cdot \vec{S}

The intensity transformation matrix $M$ is the Müller Matrix.

In order to find the Müller Matrix for an interaction, one may (if it is not stated in literature) go back to the easier accsssible transformation of the field strengths

\begin{array}{l} E_{y}^{'} = a_{1} E_{y} + a_{4} E_{z} \\ E_{z}^{'} = a_{3} E_{y} + a_{2} E_{z} \end{array}

and apply the dotted field strength to the definition of the Stokes parameters. After sorting the arising terms, elements of the Müller-Matrix become identifiable.

Example 1: Matrix of a retarder, fast axis in y direction ↓

The matrix of an ideal retarder with a phase difference $φ$ of the wave parts and a fast axis in y direction may be obtained from the transformation for the field strength:

E_{y}' = e^{- i φ / 2} E_{y} E_{z}' = e^{i φ / 2} E_{z}

Applied to the Stokes parameters one finds:

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

The transformation in matrix form:

(\begin{matrix} I_{y}' \\ I_{z}' \\ U' \\ V' \end{matrix}) = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & cos φ & - sin φ \\ 0 & 0 & sin φ & cos φ \end{matrix}) \cdot (\begin{matrix} I_{y} \\ I_{z} \\ U \\ V \end{matrix})

Example 2: Matrix of the linear polariser with diagonal orientation ↓

The transformation of the field strength for an ideal linear polariser with diagonal orientation in the first and third quadrant of the y,z plane:

\begin{array}{l} E_{y}' = \frac{1}{2} E_{y} + \frac{1}{2} E_{z} \\ E_{z}' = \frac{1}{2} E_{y} + \frac{1}{2} E_{z} \end{array}

Applied to the Stokes parameters:

\begin{array}{l} I_{y}' = E_{y}' E_{y}^{*}' = \frac{1}{4} E_{y} E_{y}^{*} + \frac{1}{4} E_{z} E_{z}^{*} + \frac{1}{4} (E_{y} E_{z}^{*} + E_{z} E_{y}^{*}) \\ = \frac{1}{4} (I_{y} + I_{z} + U) \\ I_{z}' = E_{z}' E_{z}^{*}' = \frac{1}{4} E_{y} E_{y}^{*} + \frac{1}{4} E_{z} E_{z}^{*} + \frac{1}{4} (E_{y} E_{z}^{*} + E_{z} E_{y}^{*}) \\ = \frac{1}{4} (I_{y} + I_{z} + U) \\ U' = E_{y}' E_{z}^{*}' + E_{z}' E_{y}^{*}' = \frac{1}{2} E_{y} E_{y}^{*} + \frac{1}{2} E_{z} E_{z}^{*} + \frac{1}{2} (E_{y} E_{z}^{*} + E_{z} E_{y}^{*}) \\ = \frac{1}{2} (I_{y} + I_{z} + U) \\ V' = i (E_{y}' E_{z}^{*}' - E_{z}' E_{y}^{*}') = 0 \end{array}

The transformation in matrix form:

(\begin{matrix} I_{y}' \\ I_{z}' \\ U' \\ V' \end{matrix}) = (\begin{matrix} 1 / 4 & 1 / 4 & 1 / 2 & 0 \\ 1 / 4 & 1 / 4 & 1 / 2 & 0 \\ 1 / 2 & 1 / 2 & 1 / 2 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) \cdot (\begin{matrix} I_{y} \\ I_{z} \\ U \\ V \end{matrix})

Choose examples yourself and find out to what extend the intensity and the type of polarisation of light change throughout the passage of the polarisater with this orientation!

In the way it was performed in the examples it is possible to find the Müller Matrices for the former discussed rotation of the y,z coordinates by the angle $α$ . This leads to the rotation matrix:

R (α) = (\begin{matrix} {cos}^{2} α & {sin}^{2} α & \frac{1}{2} sin 2 α & 0 \\ {sin}^{2} α & {cos}^{2} α & - \frac{1}{2} sin 2 α & 0 \\ - sin 2 α & sin 2 α & cos 2 α & 0 \\ 0 & 0 & 0 & 1 \end{matrix})

The matrices of some components with their basic orientation given in the right column can be transferred to an orientation at the angle $α$ :

M (α) = R (α) \cdot M \cdot R (- α)

The effect of several components in a row would be calculated in the very same manner as the Jones Matrices using the non-commutative product of each Müller Matrix, for n components:

M = M_{n} \cdot M_{n - 1} \cdot ... \cdot M_{2} \cdot M_{1}

In the last position of an optical setup, there is often a photodetector for measuring the intensity of light. The detector can be depicted as a row vector $\vec{D} = (\begin{matrix} 1 & 1 & 0 & 0 \end{matrix})$ by the help of which the sum of the first two elements of a Stokes Vector are calculated:

I_{y}' + I_{z}' = (\begin{matrix} 1 & 1 & 0 & 0 \end{matrix}) \cdot (\begin{matrix} I_{y}' \\ I_{z}' \\ U' \\ V' \end{matrix}) = (\begin{matrix} 1 & 1 & 0 & 0 \end{matrix}) \cdot M \cdot (\begin{matrix} I_{y} \\ I_{z} \\ U \\ V \end{matrix})

The row matrix of the detector can be further detailed by spectral factors in order to take into account its wavelength sensitivity and the quantum efficiency.

Müller Matrix	Component or interaction
$P = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix})$	Linear polariser transmitting along the y axis
$Q_{φ} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & cos φ & - sin φ \\ 0 & 0 & sin φ & cos φ \end{matrix})$	Retarder with a phase difference $φ$ of the wave parts, fast axis in y direction
$Q_{λ / 4} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & - 1 \\ 0 & 0 & 1 & 0 \end{matrix})$	$λ / 4$ retarder ( $λ / 4$ waveplate) with $φ = π / 2$ , fast axis in y direction
$Q_{λ / 2} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & - 1 \end{matrix})$	$λ / 2$ retarder ( $λ / 2$ waveplate) with $φ = π$ , fast axis in y direction
$T = \frac{n_{2} cos δ_{t}}{n_{1} cos δ} (\begin{matrix} t_{∥}^{2} & 0 & 0 & 0 \\ 0 & t_{⊥}^{2} & 0 & 0 \\ 0 & 0 & t_{∥} t_{⊥} & 0 \\ 0 & 0 & 0 & t_{∥} t_{⊥} \end{matrix})$	Fresnel refraction at a dielectric interface. The quantities n₁ and n₂ are the refractive indices of the incident and the refracted medium. Fresnel coefficients as given in the section about the Jones Matrices.
$X = (\begin{matrix} r_{∥}^{2} & 0 & 0 & 0 \\ 0 & r_{⊥}^{2} & 0 & 0 \\ 0 & 0 & - r_{∥} r_{⊥} & 0 \\ 0 & 0 & 0 & - r_{∥} r_{⊥} \end{matrix})$	Fresnel reflection at a dielectric interface. The negative signs of the mixed components result from the directional change of the light propagation due to reflection. Fresnel coefficients as given in the section about the Jones Matrices.
$Y = (\begin{matrix} r_{∥}^{2} & 0 & 0 & 0 \\ 0 & r_{⊥}^{2} & 0 & 0 \\ 0 & 0 & c & s \\ 0 & 0 & - s & c \end{matrix})$ with the abbreviations $c = r_{∥} r_{⊥} cos Δ$ $s = r_{∥} r_{⊥} sin Δ$	Fresnel reflection at a metallic surface. Angles of incidence from 0 to 90° cause phase angles Δ from 0 to 180° between the orthogonal partial waves. All quantities are functions of the complex refractive index m=n-in', where n is the real refractive index and n' is the extinction coefficient of the metal, both depending on the wavelength. A more detailed presentation is given in e.g. David Clarke: Stellar Photometry (Wiley-VCH, 2010), Appendix A.

Task: Light through a

λ / 4

retarder ↓

Please calculate the Müller Matrix of a $λ / 4$ retarder with the fast axis in the direction of the z coordinate.
Calculate the type of polarisation of the passing light for a $λ / 2$ retarder with the fast axis in direction of the z coordinate, for incoming light of the intensity 1 and having the following polarisations:
a) linear along y
b) linear diagonally in the first and third quadrant
c) linear along z
d) linear diagonally in the second and fourth quadrant

You can find the solutions on another page . Try to solve this problem yourself first!

Understanding Spectra from the Earth

Supplement 4.5: Polarisation of Electromagnetic Waves: Stokes Vectors and Müller Matrices (2/2)

Optical interactions as 4×4 matrix: Müller Matrices