The Stokes Matrix

Another way to describe a scatterer, which is widely used in optics is the Stokes matrix. The Stokes vector from (2.39) may be also written as [Ulaby90]

$$\vec{g} = \left[ \cal R \right] \vec{G} = \left[ \begin{array}{cc... ...ert E_v\vert^2 Vert E_h\vert^2 E_h E_v^* E_v E_h^* \end{array} \right]$$ (3.4)

Furthermore, between the scattered vectors $\vec{G}^{s}$ and the transmitted vector $\vec{G}^{t}$ exists the following relation [Ulaby92]

$$\vec{G}^{s} = \frac{1}{r^2} \left[ \cal W \right] \vec{G}^{t}$$ (3.5)

$$\left[ \begin{array}{c} {\vert E^{s}_{v}\vert}^{2} {\vert E^{s... ...}^{2} E^{s}_{h} {E^{s}_{v}}^* E^{s}_{v} {E^{s}_{h}}^* \end{array} \right] = \frac{1}{r^2} \left[ \begin{array}{cccc} S^*_{vv}S_{vv} & S^*_{vh... ...}^{2} E^{t}_{h} {E^{t}_{v}}^* E^{t}_{v} {E^{t}_{h}}^* \end{array} \right]$$

Using (2.40) and (2.41) we can now write the Stokes vector of the scattered wave $\vec{g}^{s}$ as

$$\vec{g}^{s} = \left[ \cal R \right] \vec{G}^s = \frac{1}{r^2} \le... ...\cal R \right]^{-1}\vec{g}^{t} = \frac{1}{r^2} \left[ \cal L \right]\vec{g}^{t}$$ (3.6)

where $\left[ \cal R \right]^{-1}$ is the inverse matrix of $\left[ \cal R \right]$

$$\left[ \cal R \right]^{-1} = \left[ \begin{array}{cccc} 1 & 1 & 0 & 0\\ 1 & -1 & 0 & 0\\ 0 & 0 & 1 & i\\ 0 & 0 & 1 & -i \end{array} \right]$$ (3.7)

$\left[ \cal L \right]$ is a $4 \times 4$ real matrix, describing the relation between the Stokes vector of the scattered wave and the Stokes vector of the transmitted wave. This $\left[ \cal L \right]$ matrix is called Stokes-matrix [Tsang95] or Müller-matrix [Jones89], [Schurcliff62]. It can be seen from the definition of the elements of $\left[ \cal W \right]$ in (3.5), that under the multiplication $S^*_{xy}S_{xy}$ with $x,y = h$ or v, the information about the absolute phase is lost.

The Stokes-matrix is often used in literature [Pottier92a], [Pottier92b], [Krogager93], and mentioned here for the sake of completeness, but will not be used further in this treatment.