Supplement 4.7: Rayleigh Scattering

The scattering matrix

We consider particles in a gas with sizes far below the wavelength of the electromagnetic waves used for irradiation, for example air molecules and visible light. Then the actual particle shape has no significant meaning, because the phase of the light wave has the same value over the whole particle. The electric field of the wave forces oscillations of its electrons, an induced dipole with a dipole moment is created

\vec{p} = α \vec{E}

with the electric field $\vec{E}$ at the location of the particle and the polarisability $α$ of the particle. The polarisability is given by the Clausius-Mossotti equation

α = \frac{3 ε_{o}}{N} \frac{n^{2} - 1}{n^{2} + 2}

with the dielectric constant ε_o=8.854·10^-12 A·s/(V·m), the particle density $N$ (number of particles per volume) and the refractive index $n$ of the gas.

Since the particles are assumed to be small and structureless, the polarisation of the illuminating light is preserved in the scattered light, as in the case of spherical particles, and the elements $a_{3}$ and $a_{4}$ of the Jones matrix are zero.

Equations ↓

The elements $a_{11}$ , $a_{12}$ , $a_{21}$ and $a_{22}$ respectively $i_{1}$ and $i_{2}$ can be measured directly; their product with the Stokes parameters $I_{y}$ und $I_{z}$ yields the parameters $I_{y}'$ and $I_{z}'$ measured with the detector $\vec{D} = (\begin{matrix} 1 & 1 & 0 & 0 \end{matrix})$ . All other Stokes parameters are only partially included in the intensities and must be calculated from several data sets if they are to be represented as individual quantities.

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

Understanding Spectra from the Earth

Supplement 4.7: Rayleigh Scattering

The scattering matrix