# A Brief Review of Mueller Matrix Calculations Associated with Oceanic Particles
## 2 Fundamental Concepts for Mueller Matrix Calculations
### Amplitude Scattering Matrix and Mueller Matrix  
Let the incident direction of the incoming wave be along the z-axis of the laboratory frame of reference. The incident direction and the scattered direction define a scattering plane and the incident and scattered fields can be expanded into parallel and perpendicular components with respect to the scattering plane. Consequently, the amplitude scattering matrix \(\mathbf{S}\) can be given by [16]  
\[\left( \begin{array}{c}\mathrm{E}^{\text{sca}}_{\parallel}\\ \mathrm{E}^{\text{sca}}_{\perp}\end{array} \right)=\frac{\exp(ikr-ikz)}{-ikr} \mathbf{S}\left( \begin{array}{c}\mathrm{E}^{\text{inc}}_{\parallel}\\ \mathrm{E}^{\text{inc}}_{\perp}\end{array} \right), \tag{9}\]  
where \(\mathrm{E}_{\parallel}\) and \(\mathrm{E}_{\perp}\) denote the parallel and perpendicular components of the electric field with respect to the scattering plane and \(\mathbf{S}\) is a \(2\times 2\) complex matrix. The Stokes parameters in a non-absorbing medium are defined based on the measurable quantities, which are normally expressed in terms of a four-element column vector, the Stokes vector \(\mathbf{I}\), as follows:  
\[\mathbf{I}=\left( \begin{array}{c}\mathrm{I}\\ \mathrm{Q}\\ \mathrm{U}\\ \mathrm{V}\end{array} \right)=\left( \begin{array}{c}\mathrm{E}_{\parallel} \mathrm{E}^{*}_{\parallel}+\mathrm{E}_{\perp}\mathrm{E}^{*}_{\perp}\\ \mathrm{E}_{\parallel}\mathrm{E}^{*}_{\perp}-\mathrm{E}_{\perp}\mathrm{E}^{*}_ {\perp}\\ \mathrm{E}_{\parallel}\mathrm{E}^{*}_{\perp}+\mathrm{E}_{\perp}\mathrm{E}^{*}_ {\parallel}\\ i\left(\mathrm{E}_{\parallel}\mathrm{E}^{*}_{\perp}-\mathrm{E}_{\perp} \mathrm{E}^{*}_{\parallel}\right)\end{array} \right). \tag{10}\]  
In the above equation, \(i\) is the imaginary unit and a constant factor, \(\sqrt{\varepsilon/u_{0}}/2\), is neglected since usually relative quantities are measured. The Mueller matrix (also called the scattering phase matrix in the literature) is the transformation matrix from the incident to the scattered Stokes parameters, as follows:  
\[\mathbf{I}^{\text{sca}}\ =\frac{1}{\left(kr\right)^{2}}\mathbf{PI}^{\text{inc}}, \tag{11}\]  
Figure 1: Parameters used for light scattering by a dielectric particle. The field point \(\overrightarrow{r}\) is outside the scattering particle with wavenumber \(k\), permittivity \(\varepsilon\), and permeability \(\mu_{0}\) and the point \(\overrightarrow{r}^{\prime}\) is inside the particle with wave number \(k_{1}\), permittivity \(\varepsilon_{1}\), and permeability \(\mu_{0}\). where the 4 \(\times\) 4 Mueller matrix \(\mathbf{P}\) can be given as quadratic expressions of the amplitude scattering matrix \(\mathbf{S}\), as follows [17; 18]:  
\[\mathbf{P}=\mathbf{A}(\mathbf{S}\otimes\mathbf{S}^{*}\ )\mathbf{A}^{-1}, \tag{12}\]  
where asterisk denotes the complex conjugate and symbol \(\otimes\) denotes the tensor product, and the constant matrix \(\mathbf{A}\) is  
\[\mathbf{A}=\left( \begin{array}{cccc}1&0&0&1\\ 1&0&0&-1\\ 0&1&1&0\\ 0&i&-i&0\end{array} \right),\quad\mathbf{A}^{-1}=\frac{1}{2}\mathbf{A}^{\dagger}, \tag{13}\]  
in which the symbol \(\dagger\) (sometimes called the dagger) is composed of two operations; namely, complex conjugating (the * symbol), and then transposing the original matrix and the order of these operations is unimportant. Note that the Stokes parameters have the units of irradiance [19], and on the other hand, the corresponding radiance is invariant over distance if no scattering or absorption occurs.