## 2 Theoretical Background
### PS-DHI for Stokes Vectors or Miller Formalism
With this aim, in Figure 6, a spherical coordinate system is shown; here, the vertical and horizontal states (V,H) are positioned on the two poles, whereas the polar and azimuthal angles are given by \(2\chi_{1}\) and \(2\psi_{1}\), respectively. In accord with [PERSON]'s theory [41], the orthogonal components intensities of the object wave \(I_{O}\) are given by  
\[ \begin{array}{ll}I_{OV}&=&|O_{V}|^{2}&=&I_{O}sin^{2}(\hat{O}H/2)\\ I_{OH}&=&|O_{H}|^{2}&=&I_{O}cos^{2}(\hat{O}H/2)\\ \end{array}  \tag{7}\]  
Once the distribution of the two components of the complex amplitude of the object wave has been retrieved, the polar angle can be evaluated as  
\[2\chi_{1}\ =\ \pi/2-\hat{O}H\ =\ \pi/2-2 arctan|O_{V}|/|O_{H}| \tag{8}\]  
As previously discussed, for each polarization state \(V\) and \(H\), the phase difference between reference and the object waves contains the contributes of both the dynamic and the geometric phase, in particular,  
\[\varphi_{V}\ =\ \varphi_{Vd}+\varphi_{Vpb} \tag{9}\]  
\[\varphi_{H}\ =\ \varphi_{Hd}+\varphi_{Hpb} \tag{10}\]  
Figure 6: Theoretical model for estimating polarization state based on geometric phase (reprinted from [PERSON] et al. [47]). In Equations (9) and (10), the first term in the sum denotes the dynamic phase difference and the second term is the PB phase. For horizontal polarization state, the PB phase is numerically equivalent to half of the area of the geodesic triangle ROH (the red region in Figure 6), i.e., \(\varphi_{tPb}=\Omega/2\)[48]. Correspondingly, for vertical polarization state, the PB phase is given by \(\varphi_{Vpb}=-2\psi_{1}+\Omega/2\). So, the azimuthal angle can be evaluated as [47]  
\[2\psi_{1}\ =\ (\varphi_{H}-\varphi_{V})-(\varphi_{Hd}-\varphi_{Vd}) \tag{11}\]  
Taking advantage by DH, the two orthogonal components of the optical field can be obtained by  
\[ \begin{array}{ll}E_{V}&=\ k|O_{V}||R_{V}|e^{i\varphi_{V}}\\ E_{H}&=\ k|O_{H}||R_{H}|e^{i\varphi_{H}}\end{array}  \tag{12}\]  
where \(k\) is a constant depending on the exposure time and response of the digital camera. The holographic reconstruction approach allows evaluating the phases \(\varphi_{V}\) and \(\varphi_{H}\), as a consequence the first term in brackets on the right side of Equation (11) can be calculated. Regarding the second term in brackets, \(\Delta\varphi\)= \((\varphi_{Hd}-\varphi_{Vd})\), it is a fixed value for the adjusted PAMDH configuration and can be corrected by measuring a linearly polarized beam along 45\({}^{\circ}\) produced by a standard polarizer. Since the two reference waves are adjusted to have the same intensity, the azimuthal and polar angles can be estimated by the following relationships:  
\[ \begin{array}{ll}2\psi_{1}\ =\ arg(E_{H}/E_{V})-\Delta\varphi\\ 2\chi_{1}\ =\ \pi/2-2 arctan(|E_{V}|/|E_{H}|)\end{array}  \tag{13}\]  
Thus, the area of \(\Omega\) can be evaluated as a function of the angles \(2\psi_{1}\) and \(2\chi_{1}\)[39] and the dynamic phases \(\varphi_{Hd}\) and \(\varphi_{Vd}\) are calculated [40; 47; 48; 49]. Finally, the normalized Stokes parameters (S\({}_{1}\), S\({}_{2}\), S\({}_{3}\)) can be expressed as a function of (\(2\psi_{1}\), \(2\chi_{1}\)) with a trigonometric relationship, as shown in Figure 6 and as described by the following relations [44]:  
\[ \begin{array}{ll}S_{1}\ =\ \sin 2\chi_{1}\\ S_{2}\ =\ \cos 2\chi_{1}\cos 2\psi_{1}\\ S_{3}\ =\ \cos 2\chi_{1}\sin 2\psi_{1}\end{array}  \tag{14}\]  
The so-evaluated Stokes parameters describe the state of polarization of the object wave at the measurement area, hence they fully characterize the SoP of an arbitrarily polarized wave. Finally, it is worth noting that the signal light from some imaging techniques, such as fluorescence imaging, is spatially incoherent, and therefore requires a further challenge for holographic imaging. Moreover, the use of light sources such as lasers, with highly temporal and spatial coherence, reduces the quality of the hologram owing to speckle and spurious fringe generation, which can decrease the spatial sensitivity of the system [50; 51].