## 3 Inherent Optical Properties (IOPs)
### Scattering Phase Matrix
#### 3.2.1 Stokes Vector Representation \(\mathbf{I}_{S}=[I,Q,U,V]^{T}\)
To compute \(\mathbf{P}_{S}(\theta^{\prime},\phi^{\prime};\theta,\phi)\) given by Equation (21) we must relate the angles \(\theta^{\prime}\), \(\phi^{\prime}\), \(\theta\), and \(\phi\) on the left side with the angles \(i_{1}\), \(i_{2}\), and \(\Theta\) on the right side. Using spherical geometry, we may apply the cosine rule for \(\Theta\), \(\theta\), and \(\theta^{\prime}\) successively, in Figure 1, to obtain (\(u=\cos\theta,u^{\prime}=\cos\theta^{\prime}\)) [5]  
\[\cos\Theta = uu^{\prime}+(1-u^{2})^{1/2}(1-u^{\prime 2})^{1/2}\cos(\phi^{ \prime}-\phi) \tag{25}\] \[\cos i_{1} = \frac{-u+u^{\prime}\cos\Theta}{(1-u^{\prime 2})^{1/2}(1-\cos^{2} \Theta)^{1/2}}\] (26) \[\cos i_{2} = \frac{-u^{\prime}+u\cos\Theta}{(1-u^{2})^{1/2}(1-\cos^{2} \Theta)^{1/2}}. \tag{27}\]  
The trigonometric functions for the double angles can be obtained by using  
\[\cos 2i=2\cos^{2}i-1 \tag{28}\]  
and  
\[\sin 2i=2\sin i\cos i \tag{29}\]  
or  
\[\sin 2i = \left\{ \begin{array}{ccc}2(1-\cos^{2}i)^{1/2}\cos i&\mbox{if}&0 <\phi^{\prime}-\phi<\pi\\ -2(1-\cos^{2}i)^{1/2}\cos i&\mbox{if}&\pi<\phi^{\prime}-\phi<2\pi\end{array} \right. \tag{30}\]  
where \(i\) is \(i_{1}\) or \(i_{2}\). We now have all the information needed to compute the scattering phase matrix [see Equation (21)] as a function of the three variables \(u\), \(u^{\prime}\) and \(\phi^{\prime}-\phi\):  
\[\mathbf{P}_{S}(u^{\prime},u,\phi^{\prime}-\phi)=\mathbf{R}_{S}(-i_{2})\mathbf{ F}_{S}(\Theta)\mathbf{R}_{S}(-i_{1}).\]  
If there is no difference in azimuth (i.e. \(\phi^{\prime}-\phi=0\)), then the meridian planes of the incident and scattered beams in Figure 1 coincide with the scattering plane. Hence there is no need to rotate the reference planes (\(\mathbf{R}(-i_{2})\) and \(\mathbf{R}(-i_{1})\) both reduce to the identity matrix), so that  
\[\mathbf{P}_{S}(u^{\prime},u,0)=\mathbf{P}_{S}(u^{\prime},u,\pi)=\mathbf{F}_{S} (\Theta). \tag{31}\]  
It follows from Equation (25) that the phase matrix is invariant to three basic changes in the polar angles \(u^{\prime}\) and \(u\) and azimuthal angles \(\phi^{\prime}\) and \(\phi\) which leave the scattering angle unaltered: (i) changing the signs of \(u\) and \(u^{\prime}\) simultaneously: \(\mathbf{P}_{S}(-u^{\prime},-u,\phi^{\prime}-\phi)=\mathbf{P}_{S}(u^{\prime},u, \phi^{\prime}-\phi)\), (ii) interchange of \(u\) and \(u^{\prime}\): \(\mathbf{P}_{S}(u^{\prime},u,\phi^{\prime}-\phi)=\mathbf{P}_{S}(u,u^{\prime},\phi^{ \prime}-\phi)\) (iii) interchange of \(\phi\) and \(\phi^{\prime}\): \(\mathbf{P}_{S}(u^{\prime},u,\phi^{\prime}-\phi)=\mathbf{P}_{S}(u^{\prime},u, \phi-\phi^{\prime})\). Also, if the \(b_{2}\)-element in Equation (22) is zero, the circular polarization component decouples from the other three components. Then, the Stokes parameter \(V\) is scattered independently of the others, according to the phase function \(a_{4}(\Theta)\), and the remaining part of the scattering phase matrix referring to \(I\), \(Q\), and \(U\) becomes a \(3\times 3\) matrix:  
\[\mathbf{P}_{S}(\Theta)= \begin{bmatrix}a_{1}&b_{1}C_{1}&-b_{1}S_{1}\\ b_{1}C_{2}&C_{2}a_{2}C_{1}-S_{2}a_{3}S_{1}&-C_{2}a_{2}S_{1}-S_{2}a_{3}C_{1}\\ b_{1}S_{2}&S_{2}a_{2}C_{1}+C_{2}a_{3}S_{1}&-S_{2}a_{2}S_{1}+C_{2}a_{3}C_{1} \end{bmatrix} . \tag{32}\]  
Finally, in a plane-parallel or slab geometry, there is no azimuth-dependence for light beams traveling in directions perpendicular to the slab (either up or down). Thus, if either the incident or the scattered beam travels in a perpendicular direction, we may use the meridian plane of the other beam as a reference plane for both beams. Since this plane coincides with the scattering plane, Equation (31) applies in this situation too.