# Polarimetric Detection of Chemotherapy-Induced Cancer Cell Death
## 2 Theory
### Analysis of the Mueller Matrix  
There are different methods to carry out **MM** analysis, each one introducing a set of polarimetric magnitudes connected to some optical properties of the sample. In this work, we will only work with some of them.  
In the Polar Decomposition (MMPD) [20], the **MM** is expressed as the product of three matrices, each one connected to a different action of the system on the input light:  
\[\textbf{MM}=\textbf{M}_{\Delta}\textbf{M}_{\textbf{R}}\textbf{M}_{\textbf{D}} \tag{3}\]  
where \(\textbf{M}_{\Delta}\), \(\textbf{M}_{\textbf{R}}\), and \(\textbf{M}_{\textbf{D}}\) are referred to as the depolarization, retardance, and diattenuation matrix, respectively. From matrix \(\textbf{M}_{\textbf{D}}\), we can obtain total diattenuation \(D_{T}\) as:  
\[D_{T}=\frac{\sqrt{(m_{D12})^{2}+(m_{D13})^{2}+(m_{D14})^{2}}}{m_{D11}}\quad. \tag{4}\]  
Diattenuation can be understood as the property whereby the output intensity depends on the polarization of the incident light. Other parameters that can be obtained from this decomposition are retardance, whereby the phase of the output light depends on the incident polarization and depolarization, which is the ability to depolarize incident polarized light.  
Regarding depolarization, a Mueller matrix **MM** is pure or non-depolarizing if it does not depolarize light entering a medium, maintaining the polarization without any loss. On the other hand, a depolarizing **MM** transforms totally-polarized light into partially polarized. The depolarization index \(D_{I}\) (Equation (5)) is a way to quantify how much a **MM** depolarizes an incident polarized beam. For pure matrices \(D_{I}=0\), on the other hand, a fully-depolarizing **MM** has \(D_{I}=1\).  
\[D_{I}=1-\sqrt{\frac{\text{tr}(\textbf{MM}\,\textbf{MM}^{T})-m_{11}^{2}}{3m_{11 }^{2}}}\quad, \tag{5}\]  
where tr represents the trace and T is the transpose.  
Another procedure used in the analysis is **MM** Transformation (MMT) [21], a method that operates directly with the **MM** elements in order to obtain meaningful magnitudes related to the anisotropy degree, depolarization power, and alignment direction of the sample. For instance, the anisotropy parameter, \(A\) is defined as:  
\[A=\frac{2 bt}{b^{2}+t^{2}}\quad, \tag{6}\]  
where \(t\) and \(b\) are given by Equations (7) and (8), respectively:  
\[t=\frac{\sqrt{(m_{22}-m_{33})^{2}+(m_{23}+m_{32})^{2}}}{2}\quad, \tag{7}\]  
\[b=\frac{m_{22}+m_{33}}{2}\quad. \tag{8}\]  
Finally, another commonly-used method to extract information is the Differential Decomposition method (MMDD) [29]. The differential decomposition of the Mueller matrix is obtained from the matrix logarithm of the macroscopic Mueller matrix \(\textbf{m}=\ln(\textbf{MM})\). This matrix can be decomposed as the sum of two matrices, one containing the non-depolarizing action of the system, \(\mathbf{m_{m}}\), and the other that fully depolarizes the beam, \(\mathbf{m_{u}}\).  
The most general form of the differential Mueller matrix for a non-depolarizing medium (\(\mathbf{m_{u}}=0\)) is [30]:  
\[\mathbf{m}=\mathbf{m_{m}}+\mathbf{m_{u}}=\left[\begin{array}{cccc}\alpha&LD &LD^{\prime}&CD\\ LD&\alpha&CB&-LB^{\prime}\\ LD^{\prime}&-CB&\alpha&LB\\ CD&LB^{\prime}&-LB&\alpha\end{array}\right]\;\;, \tag{9}\]  
where \(LD\), \(LD^{\prime}\), and \(CD\) are respectively the linear dichroism in the xy and 45-145\({}^{\circ}\) axes, and the circular dichroism, xy being the coordinates chosen for the description of the field. \(LB\), \(LB^{\prime}\), and \(CB\) stand respectively for linear birefringence in the xy and 45-135\({}^{\circ}\) axes and circular birefringence. The value \(\alpha\) is the isotropic absorption coefficient. Total dichroism and total birefringence are calculated as follows:  
\[TD=\sqrt{LD^{2}+LD^{\prime 2}+CD^{2}}\;\;\;, \tag{10}\]  
\[TB=\sqrt{LB^{2}+LB^{\prime 2}+CB^{2}}\;\;\;. \tag{11}\]