# Mueller Matrix-Based Approach for the Ex Vivo Detection of Riboflavin-Treated Transparent Biotissue
## 2 Materials and Methods
### Mueller Matrix Measurement Setup  
Mueller matrix (MM) polarimetry delivers information about the polarization changing properties of the samples under study. This information can be derived from the measurement of the intensity of light which is reflected or transmitted through a sample following illumination at different polarization states. The Mueller matrix relates to the Stokes formalism used to describe the polarization state of light before and after interaction with the sample [32; 33; 34; 35]. The Stokes vector \(\vec{S}_{\circ}\) of the light field after interaction with a sample can be calculated from the Stokes vector of the incoming light \(\vec{S}_{i}\) for known MM as follows:  
\[\left(\begin{array}{c}S_{\circ 0}\\ S_{\circ 1}\\ S_{\circ 2}\\ S_{\  
of known contributions is usually performed. Here, we employed the polar decomposition by [PERSON], which was widely used in the literature as [37]:  
\[M_{exp}=M_{\Delta}\cdot M_{R}\cdot M_{D} \tag{3}\]  
\(M_{exp}\) represents the experimentally measured MM, \(M_{\Delta}\) a depolarizer, \(M_{R}\) a retarder and \(M_{D}\) a diattenuator. For a pure depolarizer, \(M_{\Delta}\) can be written in the matrix form as  
\[M_{\Delta}= \begin{bmatrix}1&0&0&0\\ 0&a&0&0\\ 0&0&b&0\\ 0&0&0&c\end{bmatrix}  \tag{4}\]  
Further to that, key figures are the parameter \(\Delta\), the depolarization power; \(R\), the total retardance (combination of the effect of linear and circular birefringence), and \(P\), the total polarizance. These parameters can be calculated to determine specific sample properties by Equations (5)-(7) [37] as follows:  
\[\Delta=1-\frac{|a|+|b|+|c|}{3} \tag{5}\]  
\[R=cos^{-1}\Big{(}\frac{tr(M_{R})}{2}-1\Big{)} \tag{6}\]  
\[P=\frac{1}{M_{11}}\sqrt{M_{21}^{2}+M_{31}^{2}+M_{41}^{2}} \tag{7}\]  
For the measurements reported in this work, we used our in-house polarimetry system which was described in detail in [20]. A sketch of the system is shown in Figure 1.  
Here, we briefly summarize the main aspects of the system and methodology. The MM measurement system generates the polarization states of the incoming light using a polarization state generator (PSG) and records the outgoing states upon the interaction with the samples and passing through a polarization state analyzer (PSA). Using a CCD camera (BFS-U3-3254M-C, FLIR Integrated Imaging Solutions Inc., Richmond, BC, Canada), 2D measurements of the MM of a sample were obtained in one shot [20]. The PSG and PSA components were assembled from fixed polarizers (LPVISE100-A2\", Thorlabs, Newton, NJ, USA) and liquid crystal retarders (LCRs) (LCC1223T, Thorlabs, Newton, NJ, USA). By changing the voltage applied to the LCR, all required polarization states can be generated in approximately 15 s in total. The system was designed to operate in reflection and transmission mode and uses laser light at a wavelength of 532 nm (CW532-04-1, Roithner  
Figure 1: Sketch of the Mueller matrix polarimetry system. The light from the laser passes through the polarization state generator (PSG) which consists of two liquid crystal retarders (LCR) and a linear polarizer. After the interaction with the sample, the polarization states were analyzed by the polarization state analyzer (PSA) which also consists of LCRs and a linear polarizer. The LCR pairs are placed at different angles with respect to their principal axes. Lasertechnik GmbH, Vienna, Austria) for the measurement. From the measured intensity values, the MM matrix elements were calculated for each pixel, as can be seen in Table 1.  
The setup was calibrated to ensure the optical system is adjusted and behaves similarly to an ideal system. This requires the LCRs to be oriented at the correct angle with respect to each another. For this, the retardation to be set at the 532 nm wavelength needs to be determined. The detailed calibration procedure and steps are described in refs. [38; 39]. Here, we summarized the main aspects in brief.