## 3 Inherent Optical Properties (IOPs)
### Scattering Phase Matrix
#### 3.2.1 Stokes Vector Representation \(\mathbf{I}_{S}=[I,Q,U,V]^{T}\)  
The scattering geometry is illustrated in Figure 1. The plane \(\mathbf{AOB}\), defined as the scattering plane, is spanned by the directions of propagation of the incident parallel beam with Stokes vector \(\mathbf{I}_{S}^{\text{inc}}\) and the scattered parallel beam with Stokes vector \(\mathbf{I}_{S}^{\text{sca}}\). Here the subscript \(S\) pertains to the Stokes vector representation \(\mathbf{I}_{S}=[I,Q,U,V]^{T}\). The scattered radiation, represented by the Stokes vector \(\mathbf{I}_{S}^{\text{sca}}\), is related to the incident radiation, represented by the Stokes vector \(\mathbf{I}_{S}^{\text{inc}}\), by a 4\(\times\)4 scattering matrix [see Equations (18) and (19) below] and two rotations are required to properly connect the two Stokes vectors as explained below. We describe the Stokes vector of the incident beam in terms of two unit vectors \(\hat{\ell}^{\prime}\) and \(\hat{\mathbf{r}}^{\prime}\), which are normal to one another and to the unit vector \(\hat{\boldsymbol{\Omega}}^{\prime}=\hat{\mathbf{r}}^{\prime}\times\hat{\ell}^ {\prime}\) along the propagation direction of the incident beam. Similarly, we describe the Stokes vector of the scattered beam in terms of two unit vectors \(\hat{\ell}\) and \(\hat{\mathbf{r}}\), which are normal to one another and to the unit vector \(\hat{\boldsymbol{\Omega}}=\hat{\mathbf{r}}\times\hat{\ell}\) along the propagation direction of the scattered beam. The unit vector \(\hat{\ell}^{\prime}\) is along the direction of \(\mathbf{E}_{\parallel}^{\prime}\) of the incident beam and lies in the meridian plane of that beam, which is defined as the plane \(\mathbf{OAC}\) in Figure 1. Similarly, the unit vector \(\hat{\ell}\) is along the direction of \(\mathbf{E}_{\parallel}\) of the scattered beam and lies in the meridian plane of that beam, which is defined as the plane \(\mathbf{OBC}\) in Figure 1. For the incident beam, the unit vector \(\hat{\ell}^{\prime}\), may be defined to be tangent at the point \(\mathbf{A}\) to the unit circle passing through the points \(\mathbf{A}\) and \(\mathbf{C}\) in Figure 1. For the scattered beam, the unit vector \(\hat{\ell}\) may be defined to be tangent at the point \(\mathbf{B}\) to the unit circle passing through the points \(\mathbf{B}\) and \(\mathbf{C}\) in Figure 1. For either beam, its meridian plane acts as a plane of reference for the Stokes vector, so that the point \(\mathbf{A}\) in Figure 1 is the starting point for the unit vector \(\hat{\boldsymbol{\Omega}}^{\prime}=\hat{\mathbf{r}}^{\prime}\times\hat{\ell}^ {\prime}\) along the direction of propagation of the incident beam, and the point \(\mathbf{B}\) in Figure 1 is the starting point for the unit vector \(\mathbf{\hat{\Omega}}=\mathbf{\hat{r}}\times\hat{\ell}\) along the direction of propagation of the scattered beam. As explained above, the Mueller matrix describes scattering by a single particle, and for scattering by a small volume of particles, the ensemble-averaged Mueller matrix is referred to as the Stokes scattering matrix \(\mathbf{F}_{S}\). If any of the following conditions are fulfilled [15] (i) each particle in the volume element has a plane of symmetry, and the particles are randomly oriented, (ii) each volume element contains an equal number of particles and their mirror particles in random orientation, (iii) the particles are much smaller than the wavelength of the incident light, then the Stokes scattering matrix in the \(\mathbf{I}_{S}=[I,Q,U,V]^{T}\) representation has the following form  
\[\mathbf{F}_{S}(\Theta)= \begin{bmatrix}a_{1}(\Theta)&b_{1}(\Theta)&0&0\\ b_{1}(\Theta)&a_{2}(\Theta)&0&0\\ 0&0&a_{3}(\Theta)&b_{2}(\Theta)\\ 0&0&-b_{2}(\Theta)&a_{4}(\Theta)\end{bmatrix} . \tag{18}\]  
Each of the six independent matrix elements in Equation (18) depends on the scattering angle \(\Theta\), and will in general also depend on the position in the medium.