## 3 Inherent Optical Properties (IOPs)
### Scattering Phase Matrix
#### 3.2.1 Stokes Vector Representation \(\mathbf{I}_{S}=[I,Q,U,V]^{T}\)
For spherical particles, the matrix in Equation (18) simplifies, since \(a_{1}=a_{2}\) and \(a_{3}=a_{4}\), so that only four independent elements remain. As already mentioned, two rotations are required to connect the Stokes vector of the scattered radiation to that of the incident radiation. As illustrated in Figure 1, the first rotation is from the meridian plane \(\mathbf{OAC}\), associated with the Stokes vector \(\mathbf{I}_{S}^{\text{inc}}\), into the scattering plane \(\mathbf{OAB}\), whereas the second rotation is from the scattering plane \(\mathbf{OAB}\) into the meridian plane \(\mathbf{OBC}\), associated with the Stokes vector \(\mathbf{I}_{S}^{\text{sca}}\). Hence, the Stokes vector for the scattered radiation is given by [16]  
\[\mathbf{I}_{S}^{\text{sca}}=\mathbf{R}_{S}(\pi-i_{2})\mathbf{F}_{S}(\Theta) \mathbf{R}_{S}(-i_{1})\mathbf{I}_{S}^{\text{inc}}\equiv\mathbf{P}_{S}(\Theta )\mathbf{I}_{S}^{\text{inc}}. \tag{19}\]  
The matrix \(\mathbf{R}_{S}\) is called the Stokes rotation matrix. It represents a rotation in the clockwise direction with respect to an observer looking into the direction of propagation, and can be written as (\(0\leq\omega\leq 2\pi\))  
\[\mathbf{R}_{S}(\omega)= \begin{bmatrix}1&0&0&0\\ 0&\cos(2\omega)&-\sin(2\omega)&0\\ 0&\sin(2\omega)&\cos(2\omega)&0\\ 0&0&0&1\end{bmatrix} . \tag{20}\]  
Hence, according to Equation (19), the scattering phase matrix, which connects the Stokes vector of the scattered radiation to that of the incident radiation, is obtained from the Stokes scattering matrix \(\mathbf{F}_{S}(\Theta)\) in Equation (18) by  
\[\mathbf{P}_{S}(\theta^{\prime},\phi^{\prime};\theta,\phi)=\mathbf{R}_{S}(\pi -i_{2})\mathbf{F}_{S}(\Theta)\mathbf{R}_{S}(-i_{1})=\mathbf{R}_{S}(-i_{2}) \mathbf{F}_{S}(\Theta)\mathbf{R}_{S}(-i_{1}) \tag{21}\]  
where \(\mathbf{R}_{S}\) is the rotation matrix described in Equation (20) [16], and \(\mathbf{R}_{S}(\pi-i_{2})=\mathbf{R}_{S}(-i_{2})\) since the rotation matrix is periodic with a period \(\pi\). According to Equation (19) (see also Figure 1), the Stokes vector \(\mathbf{I}_{S}^{\text{inc}}\) of the incident parallel beam must be multiplied by the rotation matrix \(\mathbf{R}_{S}(-i_{1})\) before it is multiplied by the Stokes scattering matrix \(\mathbf{F}_{S}(\Theta)\), whereafter it must be multiplied by the rotation matrix \(\mathbf{R}_{S}(\pi-i_{2})\). These matrix multiplications are carried out explicitly in some radiative transfer (RT) models including Monte Carlo simulations, while they are implicitly taken care of in other RT models such as the adding-doubling method [17] and the discrete ordinate method [18; 19] which use the expansion of the scattering phase matrix in generalized spherical functions [20; 21] as discussed in Section 3.2.3. Carrying out the matrix multiplications in Equation (21) one finds:\[\mathbf{P}_{S}(\Theta)= \begin{bmatrix}a_{1}&b_{1}C_{1}&-b_{1}S_{1}&0\\ b_{1}C_{2}&C_{2}a_{2}C_{1}-S_{2}a_{3}S_{1}&-C_{2}a_{2}S_{1}-S_{2}a_{3}C_{1}&-b_{2} S_{2}\\ b_{1}S_{2}&S_{2}a_{2}C_{1}+C_{2}a_{3}S_{1}&-S_{2}a_{2}S_{1}+C_{2}a_{3}C_{1}&-b_{2} C_{2}\\ 0&-b_{2}S_{1}&-b_{2}C_{1}&a_{4}\end{bmatrix}  \tag{22}\]  
where \(a_{i}=a_{i}(\Theta),i=1,\ldots,4\), \(b_{i}=b_{i}(\Theta),i=1,2\), and  
\[C_{1} = \cos 2i_{1},\qquad C_{2}=\cos 2i_{2} \tag{23}\] \[S_{1} = \sin 2i_{1},\qquad S_{2}=\sin 2i_{2}. \tag{24}\]  
A comparison of Equations (18) and (22) shows that only the corner elements of \(\mathbf{F}_{S}(\Theta)\) remain unchanged by the rotations of the reference planes. The (1,1)-element of the scattering phase matrix \(\mathbf{P}_{S}(\Theta)\) (and of the Stokes scattering matrix \(\mathbf{F}_{S}(\Theta)\)) is the scattering phase function. Since also the (4,4)-element of the scattering phase matrix remains unchanged by the rotations, the state of circular polarization of the incident light does not affect the intensity of the scattered radiation after one scattering event.