## 2 Materials and Methods
### Birefringence Physical Model  
Birefringent materials are characterized by optical anisotropy, since their refractive index varies depending on the direction of incident light with respect to the material main optical axes. The anisotropy of the refractive index can be represented by an ellipsoid named indicatrix. The intersection of the indicatrix with the plane perpendicular to the incident light is an elliptical section, which defines two polarization axes: one where the refractive index is larger (slow axis) and one where it is smaller (fast axis). The components of the electrical field along these axes undergo a relative phase shift:  
\[\delta=\frac{2\pi}{\lambda}\Delta n\cdot L \tag{1}\]  
where \(\Delta n\) is the difference of the refractive indices and \(L\) is the light path, i.e., the sample thickness.  
A common configuration to study the birefringence effect in microscopy is to place the sample between two linear polarizers (polarizer-sample-analyzer configuration, LP-S-LP). The polarized light emerging from the sample has its components along the birefringent axes with a phase shift \(\delta\). When the projections of these components on the transmission axis of the analyzer recombine, the phase shift causes an interference, which modulates the intensity of the light emerging from the analyzer. The light intensity collected from an LP-S-LP configuration for ideal polarizers is computed as in [20]:  
\[I=\frac{I_{o}}{2}T_{s}\bigg{[}\cos^{2}\chi-\sin 2\phi\sin 2(\phi-\chi)\sin^{2} \frac{\delta}{2}\bigg{]} \tag{2}\]  
where \(I_{o}\) is the intensity of the source unpolarized light, \(T_{s}\) is the sample transmittance, \(\chi\) is the angle between the polarizer and analyzer transmission axes, \(\phi\) is the angle between the sample slow axis and the polarizer transmission axis. A Beer-Lambert-Bouguer linear attenuation model with attenuation coefficient \(\mu\) can be assumed for the sample transmittance:  
\[T_{s}=e^{-\mu L} \tag{3}\]  
However, real polarizers are characterized by a transmittance along the transmitting axis \(T_{1}\) smaller than unity and a transmittance along the extinction axis \(T_{2}\) larger than zero [21]. The light contribution transmitted along any of the four possible combinations of principal axes of the analyzer and polarizer can be obtained from Equation (2) changing the angles \(\chi\) and \(\phi\) by \(\frac{\pi}{2}\) accordingly. The total light intensity is then computed as:  
\[\begin{array}{ll}I=\frac{I_{o}}{2}T_{s}\big{[}\big{(}T_{1}{}^{2}+&T_{2}{}^{2 }\big{)}\cos^{2}\chi+2T_{1}T_{2}\sin^{2}\chi\\ &-\big{(}T_{1}{}^{2}+T_{2}{}^{2}-2T_{1}T_{2}\big{)}\sin 2\phi\sin 2(\phi-\chi) \sin^{2}\frac{\delta}{2}\big{]}\\ &=I_{o}T_{s}\big{[}T_{p}\cos^{2}\chi+T_{c}\sin^{2}\chi\\ &-\big{(}T_{p}-T_{c}\big{)}\sin 2\phi\sin 2(\phi-\chi)\sin^{2}\frac{\delta}{2} \Big{]}\end{array} \tag{4}\]  
where \(T_{p}\) and \(T_{c}\) are the transmittances of the polarizer pair in parallel and crossed configurations, respectively. In this model, \(T_{p}\) and \(T_{c}\) are considered images that include the inhomogeneous field illumination caused by the microscope filters and optics, while the light source \(I_{o}\) is a scalar.  
When the polarizer and the analyzer are parallel (\(\chi=0\)) Equation (4) reduces to:  
\[I_{p}=I_{o}T_{s}\bigg{[}T_{p}-\big{(}T_{p}-T_{c}\big{)}\sin^{2}2\phi\sin^{2} \frac{\delta}{2}\bigg{]}=I_{o}T_{s}\bigg{[}T_{p}-\big{(}T_{p}-T_{c}\big{)}K_{ s}\sin^{2}\frac{\delta}{2}\bigg{]} \tag{5}\]where the effect of \(\sin^{2}2\phi\) in a biological sample is expressed through the modulation image \(K_{s}\leq 1\), which is dependent on the local prevalent direction of the birefringent fibers in the tissue.  
Similarly, when the polarizer and the analyzer are perpendicular (\(\chi=\frac{\pi}{2}\)), Equation (4) reduces to:  
\[I_{c}=I_{0}T_{s}\bigg{[}T_{c}+\big{(}T_{p}-T_{c}\big{)}K_{s}\sin^{2}\frac{ \delta}{2}\bigg{]} \tag{6}\]