## 3 Inherent Optical Properties (IOPs)
### Scattering Phase Matrix
#### 3.2.3 Generalized Spherical Functions  
The \"Greek Constants\"  
For unpolarized radiation, only the \(a_{1}(\Theta)\) element of the Stokes scattering matrix Equation (18) is relevant, and this element is the scattering phase function given by Equation (5) in general, and by Equation (9) for Rayleigh scattering. As discussed below, the scattering phase function can be expanded in Legendre polynomials (see Equation (116)), enabling one to express it as a Fourier cosine series (see Equation (115)). In a similar manner, the scattering phase matrix can be expanded in generalized spherical functions. In the Stokes vector representation \(\mathbf{I}_{S}=[I,Q,U,V]^{T}\), the scattering phase matrix is \(\mathbf{P}_{S}(\Theta)=\mathbf{P}_{S}(u^{\prime},u;\phi^{\prime}-\phi)\) with \(u=\cos\theta\), \(\theta\) being the polar angle after scattering, and \(u^{\prime}=\cos\theta^{\prime}\), \(\theta^{\prime}\) being the polar angle prior to scattering. Similarly, \(\phi\) and \(\phi^{\prime}\) are the azimuth angles after and prior to scattering, respectively. To accomplish the expansion in generalized spherical functions, the scattering phase matrix is first expanded in a \((M+1)\)-term Fourier series in the azimuth angle difference (\(\Delta\phi=\phi^{\prime}-\phi\)):\[\mathbf{P}_{S}(u^{\prime},u;\Delta\phi)=\sum_{m=0}^{M}\{\mathbf{P}_{c}^{m}(u^{ \prime},u)\text{cos }m(\Delta\phi)+\mathbf{P}_{s}^{m}(u^{\prime},u)\text{sin }m(\Delta\phi)\} \tag{44}\]  
where \(\mathbf{P}_{c}^{m}(u^{\prime},u)\) and \(\mathbf{P}_{s}^{m}(u^{\prime},u)\) are the coefficient matrices of the cosine and sine terms, respectively, of the Fourier series. An addition theorem for the generalized spherical functions can be used to express the Fourier expansion coefficient matrices directly in terms of the expansion coefficients of the Stokes scattering matrix \(\mathbf{F}_{S}(\Theta)\) [see Equation (18)] as follows [20; 21; 23]:  
\[\mathbf{P}_{c}^{m}(u^{\prime},u) = \mathbf{A}^{m}(u^{\prime},u)+\boldsymbol{\Delta}_{3A}\mathbf{A}^{ m}(u^{\prime},u)\boldsymbol{\Delta}_{3,4} \tag{45}\] \[\mathbf{P}_{s}^{m}(u^{\prime},u) = \mathbf{A}^{m}(u^{\prime},u)\boldsymbol{\Delta}_{3,4}- \boldsymbol{\Delta}_{3,4}\mathbf{A}^{m}(u^{\prime},u) \tag{46}\]  
where \(\boldsymbol{\Delta}_{3}=\text{diag}(1,1,-1,1)\). The matrix \(\mathbf{A}^{m}(u^{\prime},u)\) is given by:  
\[\mathbf{A}^{m}(u^{\prime},u)=\sum_{\ell=m}^{M}\mathbf{P}_{\ell}^{m}(u) \boldsymbol{\Lambda}_{\ell}\mathbf{P}_{\ell}^{m}(u^{\prime}) \tag{47}\]  
where  
\[\boldsymbol{\Lambda}_{\ell} = \left( \begin{array}{cccc}\alpha_{1,\ell}&\beta_{1,\ell}&0&0\\ \beta_{1,\ell}&\alpha_{2,\ell}&0&0\\ 0&0&\alpha_{3,\ell}&\beta_{2,\ell}\\ 0&0&-\beta_{2,\ell}&\alpha_{4,\ell}\end{array} \right) \tag{48}\]  
and  
\[a_{1}(\Theta) = \sum_{\ell=0}^{M}\alpha_{1,\ell}P_{\ell}^{0,0}(\cos\Theta) \tag{49}\] \[a_{2}(\Theta)+a_{3}(\Theta) = \sum_{\ell=2}^{M}(\alpha_{2,\ell}+\alpha_{3,\ell})P_{\ell}^{2,2}( \cos\Theta)\] (50) \[a_{2}(\Theta)-a_{3}(\Theta) = \sum_{\ell=2}^{M}(\alpha_{2,\ell}-\alpha_{3,\ell})P_{\ell}^{2-2}( \cos\Theta)\] (51) \[a_{4}(\Theta) = \sum_{\ell=0}^{M}\alpha_{4,\ell}P_{\ell}^{0,0}(\cos\Theta)\] (52) \[b_{1}(\Theta) = \sum_{\ell=2}^{M}\beta_{1,\ell}P_{\ell}^{0,2}(\cos\Theta)\] (53) \[b_{2}(\Theta) = \sum_{\ell=2}^{M}\beta_{2,\ell}P_{\ell}^{0,2}(\cos\Theta). \tag{54}\]  
Here the so-called \"Greek constants\" \(\alpha_{j,\ell}\) and \(\beta_{j,\ell}\) are expansion coefficients, and \(a_{j}(\Theta)\) and \(b_{j}(\Theta)\) are the elements of the Stokes scattering matrix \(\mathbf{F}_{S}(\Theta)\) in Equation (18). An example of Greek constants for Rayleigh scattering is provided in Table 1 (see [24]): \begin{table}
\begin{tabular}{c c c c c c c} \hline \(\boldsymbol{\ell}\) & \(\boldsymbol{\alpha}_{1,\ell}\) & \(\boldsymbol{\alpha}_{2,\ell}\) & \(\boldsymbol{\alpha}_{3,\ell}\) & \(\boldsymbol{\alpha}_{4,\ell}\) & \(\boldsymbol{\beta}_{1,\ell}\) & \(\boldsymbol{\beta}_{2,\ell}\) \\ \hline
1 & 0 & 0 & 0 & \(3d/2\) & 0 & 0 \\
2 & \(c/2\) & \(3c\) & 0 & 0 & \(\sqrt{3}/2c\) & 0 \\ \hline \end{tabular}
\end{table}