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The Eq. 1 in Scattering matrix can be extended to a dispersion with many particles. Indeed, for a volume, dV (Fig. 1), of a dispersion with dN = n dV particles, where n is the number of particles per unit volume, we have:
| Ss(θ) = [n dV / (kr)2] M(θ) Si | (1) |
Each element, S, of the scattered and incident Stokes vectors, Ss and Si respectively, has a dimension of irradiance. Hence, in reference to Fig. 1, we have
| S | = dΦ / dAΦ | |
| = dΦ / (dω r2) | ||
| = I / r2 | (2) |
where I is the intensity of light.
Fig. 1. Scattering geometry for a dispersion: a face, with an area dA, of an elementary volume dV of the dispersion, containing N particles with identical optical properties, is illuminated by an electromagnetic radiation (ER, light) beam with irradiance E. At a scattering angle θ the elementary volume scatters power dΦ within an elementary solid angle dω that at a distance r subtends an area dAΦ.
Thus, Eq. 1 can be rewritten as follows:
| Is(θ) = [n / k2 M(θ)] dV Si | (3) |
where Is is the Stokes vector of the scattered light expressed in the intensity units. By comparing this equation with the definition of the scattering function, β, (Eq. 2 in Scattering function), it can be seen that:
| β(θ) | = n M(θ) / k2 | (4) |
where β can be regarded a polarization-dependent scattering function, a matrix operator with 4x4 elements, i.e. an equivalent of the scattering matrix in the scattering-function formalism. Note that, in contrast to the scattering matrix, the operator β converts the irradiance of the incident beam to the intensity of the scattered light.
The element β11 of the β matrix (that we denote simply by β following the traditional notation) represents the scattering properties of a dispersion for unpolarized light. It is a counterpart of the element M11 of the scattering matrix, M, by virtue of Eq. 2 of Scattering matrix for a dispersion.
| CITATION: Jonasz M. 2006. Relationship between the scattering matrix and function (www.tpdsci.com/Tpc/ScaMtxFnCnv.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 24-Mar-2006 Modified: 19-Sep-2006 Peer-reviewed: PENDING |
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