Supplement 4.6: The Scattering Matrix (2/2)

Experimental setups

The experimental examination of the scattering matrix elements requires a light source emitting a collimated beam as well as linear polarisers and λ/4-retarders

for the creation of linearly and circularly polarised light to illuminate the particles
and for the analysis of the state of polarisation of the scattering light

as well as a photodetector $\vec{D} = (\begin{matrix} 1 & 1 & 0 & 0 \end{matrix})$ with a narrow detection angle for measuring the intensities of the scattering light at defined scattering angles.

Equations ↓

The elements $a_{11}$ , $a_{12}$ , $a_{21}$ and $a_{22}$ respectively $i_{1}$ and $i_{2}$ can be directly measured; multiplying them with the Stokes-Parameters $I_{y}$ and $I_{z}$ results in the parameters $I_{y}'$ and $I_{z}'$ measured by the detector.

All other elements of the scattering matrix are only partially incorporated in the directly measurable intensities and would have to be calculated from data of several measurements if desired to be displayed as individual quantities.

Coefficient of the scattering matrix	Optical measuring assembly	Results of the measurement of particles of any shape	Results of the measurement of spherically shaped particles
$a_{11}$	$D \cdot P (0) \cdot S \cdot ℓ_{0}$	$a_{11}$	$i_{1} / k^{2}$
$a_{12}$	$D \cdot P (0) \cdot S \cdot ℓ_{90}$	$a_{12}$	0
$a_{13}$	$D \cdot P (0) \cdot S \cdot ℓ_{45}$	$\frac{1}{2} (a_{11} + a_{12}) + a_{13}$	$i_{1} / 2 k^{2}$
$a_{14}$	$D \cdot P (0) \cdot S \cdot c_{r}$	$\frac{1}{2} (a_{11} + a_{12}) + a_{14}$	$i_{1} / 2 k^{2}$
$a_{21}$	$D \cdot P (90) \cdot S \cdot ℓ_{0}$	$a_{21}$	0
$a_{22}$	$D \cdot P (90) \cdot S \cdot ℓ_{90}$	$a_{22}$	$i_{2} / 2 k^{2}$
$a_{23}$	$D \cdot P (90) \cdot S \cdot ℓ_{45}$	$\frac{1}{2} (a_{21} + a_{22}) + a_{23}$	$i_{2} / 2 k^{2}$
$a_{24}$	$D \cdot P (90) \cdot S \cdot c_{r}$	$\frac{1}{2} (a_{21} + a_{22}) + a_{24}$	$\frac{1}{2 k^{2}} i_{2}$
$a_{31}$	$D \cdot P (45) \cdot S \cdot ℓ_{0}$	$\frac{1}{2} (a_{11} + a_{21} + a_{31})$	$i_{1} / 2 k^{2}$
$a_{32}$	$D \cdot P (45) \cdot S \cdot ℓ_{90}$	$\frac{1}{2} (a_{12} + a_{22} + a_{32})$	$i_{2} / 2 k^{2}$
$a_{33}$	$D \cdot P (45) \cdot S \cdot ℓ_{45}$	$\frac{1}{4} (a_{11} + a_{12} + 2 a_{13} + a_{21} + a_{22} + 2 a_{23} + a_{31} + a_{32} + 2 a_{33})$	$(i_{1} + i_{2} + 2 i_{3}) / 4 k^{2}$
$a_{34}$	$D \cdot P (45) \cdot S \cdot c_{r}$	$\frac{1}{4} (a_{11} + a_{12} + 2 a_{14} + a_{21} + a_{22} + 2 a_{24} + a_{31} + a_{32} + 2 a_{34})$	$(i_{1} + i_{2} + 2 i_{4}) / 4 k^{2}$
$a_{41}$	$D \cdot P (0) \cdot Q_{λ / 4} (45) \cdot S \cdot ℓ_{0}$	$\frac{1}{2} (a_{11} + a_{21} + a_{41})$	$i_{1} / 2 k^{2}$
$a_{42}$	$D \cdot P (0) \cdot Q_{λ / 4} (45) \cdot S \cdot ℓ_{90}$	$\frac{1}{2} (a_{12} + a_{22} + a_{42})$	$i_{2} / 2 k^{2}$
$a_{43}$	$D \cdot P (0) \cdot Q_{λ / 4} (45) \cdot S \cdot ℓ_{45}$	$\frac{1}{4} (a_{11} + a_{12} + 2 a_{13} + a_{21} + a_{22} + 2 a_{23} + a_{41} + a_{42} + 2 a_{43})$	$(i_{1} + i_{2} - 2 i_{4}) / 4 k^{2}$
$a_{44}$	$D \cdot P (0) \cdot Q_{λ / 4} (45) \cdot S \cdot c_{r}$	$\frac{1}{4} (a_{11} + a_{12} + 2 a_{14} + a_{21} + a_{22} + 2 a_{24} + a_{41} + a_{42} + 2 a_{44})$	$(i_{1} + i_{2} + 2 i_{3}) / 4 k^{2}$
	$D \cdot S \cdot r$	$\frac{1}{2} (a_{11} + a_{12} + a_{21} + a_{22})$	$(i_{1} + i_{2}) / 2 k^{2}$

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