Supplement 4.6: The Scattering Matrix (1/2)

Scattering of light is defined as the change of the light's direction of propagation after hitting particles. It is dependent on a large variety of parameters. What material are the particles from? What is the refractive index of the material? Does it absorb the light or is it transparent? The particles' structure, shape and size are important as well as their ordered or statistically randomised position. Likewise, the wavelength of light and its polarisation are of importance.

These properties determine the intensity of scattering and its angular distribution. In addition, the properties of the medium (air, water, ...) in which the particles are suspended are essential: is the medium homogeneous and isotropic or does it have spatial structures such as crystals do?

For a basic understanding of light scattering, it is sufficient to assume simplified conditions. We therefore assume an isotropic medium and that the particles are randomly distributed in space. As a second step we will add on the assumption of spherical particles. Thus, calculating the scattering of light is not too complicated.

Scattering by randomly distributed particles

Intensities and polarisation of the scattered light are displayed in the scattering matrix $S$ which is an example of the Müller Matrices described in the previous section. Regardless of any simplifying assumptions, it consists of 4×4 independent elements $a_{11} ... a_{44}$ :

(\begin{matrix} I_{y}' \\ I_{z}' \\ U' \\ V' \end{matrix}) = (\begin{matrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{matrix}) \cdot (\begin{matrix} I_{y} \\ I_{z} \\ U \\ V \end{matrix}) \Leftrightarrow \vec{S}' = S \cdot \vec{S}

The Stokes Vector $\vec{S}$ indicates the illuminating light, whereas the vector $\vec{S}'$ indicates the scattered light. The symbol $S$ without overarrow is the scattering matrix.

Including all 16 elements, the theoretical or experimental examination of a scattering matrix would be rather complicated. Francis Perrin showed that merely six elements remain, if the particles are randomly distributed in an isotropic medium:

S = (\begin{matrix} a_{11} & a_{12} & 0 & 0 \\ a_{12} & a_{22} & 0 & 0 \\ 0 & 0 & a_{33} & a_{34} \\ 0 & 0 & - a_{34} & a_{44} \end{matrix})

Allowing for particles of any shape, their illumination with completely polarised light may result in partially depolarised scattered light. The element $a_{12}$ generates depolarised scattered light as can be seen in the example of illuminating particles with linear polarised light along the y coordinate:

(\begin{matrix} a_{11} \\ a_{21} \\ 0 \\ 0 \end{matrix}) = (\begin{matrix} a_{11} & a_{12} & 0 & 0 \\ a_{21} & a_{22} & 0 & 0 \\ 0 & 0 & a_{33} & a_{34} \\ 0 & 0 & - a_{34} & a_{44} \end{matrix}) \cdot (\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}) = S \cdot ℓ_{o}

The scattered light may be broken down into a polarised and an unpolarised component:

(\begin{matrix} a_{11} \\ a_{12} \\ 0 \\ 0 \end{matrix}) = (\begin{matrix} a_{11} - a_{12} \\ 0 \\ 0 \\ 0 \end{matrix}) + (\begin{matrix} a_{12} \\ a_{12} \\ 0 \\ 0 \end{matrix}) = (\begin{matrix} I_{y, p} \\ 0 \\ 0 \\ 0 \end{matrix}) + (\begin{matrix} I_{y, u} \\ I_{z, u} \\ 0 \\ 0 \end{matrix})

where the indices p and u represent the polarised respectively the unpolarised parts. For the unpolarised part necessarily has to be true: $I_{y, u} = I_{z, u}$ , and $U = V = 0$ .

Equations ↓

Scattering by spherical particles

Due to spherical symmetry, which does not allow for any preferred direction, no polarisation or depolarisation of the scattering light is possible. The scattering matrix becomes:

S = (\begin{matrix} a_{11} & 0 & 0 & 0 \\ 0 & a_{22} & 0 & 0 \\ 0 & 0 & a_{33} & a_{34} \\ 0 & 0 & - a_{34} & a_{33} \end{matrix})

In this case, there are merely three variable elements, since:

a_{11} a_{22} = a_{33}^{2} + a_{34}^{2}

This accords with the statement about the scattering light being entirely polarised (the Stokes Vector of fully polarised light has only three independent Stokes parameters as well). The validity of the relation can be calculated by the help of the amplitude elements $a_{1}$ and $a_{2}$ of the Jones Matrix (the amplitude-transformation matrix):

\begin{array}{l} a_{11} = a_{1} a_{1} * \\ a_{22} = a_{2} a_{2} * \\ a_{33} = (a_{1} a_{2} * + a_{2} a_{1} *) / 2 \\ a_{34} = i (a_{1} a_{2} * - a_{2} a_{1} *) / 2 \end{array}

( $i$ stands for the complex number $\sqrt{- 1}$ and $*$ indicates the conjugated complex of each complex quantity.

Scattering by particles of differents sizes ↓

If the particles are of different size, however, it applies that:

a_{11} a_{22} \geq a_{33}^{2} + a_{34}^{2}

the scattered light is merely partially polarised when illuminated with polarised light. An exception is represented by scattering angles of 0° and 180°, where the scattered light stays entirely polarised due to the axial symmetry.

The Jones Matrix for light scattering by spherical particles is a diagonal matrix since the elements $a_{3}$ and $a_{4}$ equal zero:

(\begin{matrix} a_{1} & 0 \\ 0 & a_{2} \end{matrix})

Electromagnetic theories of light scattering aim to calculate the amplitude functions

S_{1} = k a_{1}

and

S_{2} = k a_{2}

which are affiliated with the elements $a_{1}$ and $a_{2}$ of the Jones Matrix through the wave number $k = 2 π / λ$ respectively the wavelength $λ$ .