## 3 Inherent Optical Properties (IOPs)
### Scattering Phase Matrix
#### 3.2.2 Stokes Vector Representation \(\mathbf{I}=[I_{||},I_{\perp},U,V]^{T}\)  
The Stokes vector \(\mathbf{I}=[I_{||},I_{\perp},U,V]^{T}\) is related to \(\mathbf{I}_{S}=[I,Q,U,V]^{T}\) by  
\[\mathbf{I}_{S}=\mathbf{D}\mathbf{I} \tag{33}\]  
where \(\mathbf{D}\) is given by Equation (13), so that \(I=I_{||}+I_{\perp}\), and \(Q=I_{||}-I_{\perp}\). Denoting the Stokes vector obtained after a rotation by  
\[\mathbf{I}_{S}^{\prime}=\mathbf{R}_{S}(\omega)\mathbf{I}_{S} \tag{34}\]  
we find  
\[\mathbf{I}^{\prime}=\mathbf{D}^{-1}\mathbf{I}_{S}^{\prime}=\mathbf{D}^{-1} \mathbf{R}_{S}(\omega)\mathbf{I}_{S}=\mathbf{D}^{-1}\mathbf{R}_{S}(\omega) \mathbf{D}\,\mathbf{I}=\mathbf{R}(\omega)\mathbf{I}. \tag{35}\]  
Hence, the rotation matrix for the Stokes vector in the representation \(\mathbf{I}=[I_{||},I_{\perp},U,V]^{T}\) becomes:  
\[\mathbf{R}(\omega)=\mathbf{D}^{-1}\mathbf{R}_{S}(\omega)\mathbf{D}= \begin{bmatrix} \cos^{2}\omega&\sin^{2}\omega&-\frac{1}{2}\sin(2\omega)&0\\ \sin^{2}\omega&\cos^{2}\omega&\frac{1}{2}\sin(2\omega)&0\\ \sin(2\omega)&-\sin(2\omega)&\cos(2\omega)&0\\ 0&0&0&1\end{bmatrix} . \tag{36}\]  
The scattering phase matrix \(\mathbf{P}(\Theta)\) in the Stokes vector representation \(\mathbf{I}=[I_{||},I_{\perp},U,V]^{T}\) is related to scattering phase matrix \(\mathbf{P}_{S}(\Theta)\) in the Stokes vector representation \(\mathbf{I}_{S}=[I,Q,U,V]^{T}\) by  
\[\mathbf{P}(\Theta)=\mathbf{D}^{-1}\mathbf{P}_{S}(\Theta)\mathbf{D}. \tag{37}\]  
Similarly, the Stokes scattering matrix \(\mathbf{F}(\Theta)\) associated with the Stokes vector representation \(\mathbf{I}=[I_{||},I_{\perp},U,V]^{T}\) is related to the Stokes scattering matrix \(\mathbf{F}_{S}(\Theta)\) in Equation (18) by  
\[\mathbf{F}(\Theta)=\mathbf{D}^{-1}\mathbf{F}_{S}(\Theta)\mathbf{D}= \begin{pmatrix} \frac{1}{2}(a_{1}+a_{2}+2b_{1})&\frac{1}{2}(a_{1}-a_{2})&0&0\\ \frac{1}{2}(a_{1}-a_{2})&\frac{1}{2}(a_{1}+a_{2}-2b_{1})&0&0\\ 0&0&a_{3}&b_{2}\\ 0&0&-b_{2}&a_{4}\end{pmatrix} . \tag{38}\]  
For Rayleigh scattering with parameter \(f=\frac{1-\rho}{1+\rho}\), where \(\rho\) is the depolarization factor defined in Equation (43), the Stokes scattering matrix in the Stokes vector representation \(\mathbf{I}_{S}=[I,Q,U,V]^{T}\) is given by [16,22]\[\mathbf{F}_{S}(\Theta)=\frac{3}{3+f} \begin{bmatrix}1+f\cos^{2}\Theta&-f\sin^{2} \Theta&0&0\\ -f\sin^{2}\Theta&f(1+\cos^{2}\Theta)&0&0\\ 0&0&2f\cos\Theta&0\\ 0&0&0&(3f-1)\cos\Theta\end{bmatrix} . \tag{39}\]  
For the first scattering event of unpolarized light, only the (1,1)-element of Equation (39) matters, and leads to the scattering phase function given by Equation (9). In the Stokes vector representation \(\mathbf{I}=[I_{||},I_{\perp},U,V]^{T}\), the corresponding Stokes scattering matrix for Rayleigh scattering becomes (using Equations (38) and (39) [16]):  
\[\mathbf{F}(\Theta)=\frac{3}{2(1+2\zeta)} \begin{pmatrix}\cos^{2}\Theta+\zeta \sin^{2}\Theta&\zeta&0&0\\ \zeta&1&0&0\\ 0&0&(1-\zeta)\cos\Theta&0\\ 0&0&0&(1-3\zeta)\cos\Theta\end{pmatrix}  \tag{40}\]  
where \(\zeta=\rho/(2-\rho)=\frac{1-f}{1+3f}\). From Equation (40) we see that for an incident beam of natural unpolarized light given by \(\mathbf{I}^{\mathrm{inc}}=[I_{\parallel}^{\mathrm{inc}},I_{\perp}^{\mathrm{ inc}},U^{\mathrm{inc}},V^{\mathrm{inc}}]^{T}=[\frac{1}{2}I^{\mathrm{inc}}, \frac{1}{2}I^{\mathrm{inc}},0,0]^{T}\), the scattered intensities in the plane parallel and perpendicular to the scattering plane are obtained by carrying out the multiplication \(\mathbf{I}^{\mathrm{sca}}=\mathbf{F}(\Theta)\mathbf{I}^{\mathrm{inc}}\):  
\[I_{\parallel}^{\mathrm{sca}} \propto \frac{3}{4(1+2\zeta)}[2\zeta+(1-\zeta)\cos^{2}\Theta]I^{\mathrm{ inc}} \tag{41}\] \[I_{\perp}^{\mathrm{sca}} \propto \frac{3}{4(1+2\zeta)}[(1+\zeta)]I^{\mathrm{inc}}. \tag{42}\]  
Thus, for unpolarized incident light, the scattered light at right angles (\(\Theta=90^{\circ}\)) to the direction of incidence defines the depolarization ratio:  
\[\rho\equiv \begin{pmatrix}I_{\parallel}^{\mathrm{sca}}\\ \hline I_{\perp}^{\mathrm{sca}}\end{pmatrix}