## 2 Theoretical Background
### PS-DHI for Jones Formalism
Figure 2: (**a**) Example of polarization hologram; in the inset the two fringe patterns are highlighted; (**b**) Fourier amplitude spectra of the hologram: the frequencies of the zero-order of diffraction, of the virtual and real images, and of parasitic interferences (P) are clearly visible. Since \(R_{1}\) and \(R_{2}\) are orthogonally polarized, they do not interfere: \(R_{1}\):\(R_{2}\)* = \(R_{1}\)*\(R_{2}\) = 0 (where the asterisk indicates the complex conjugate), so the hologram intensity at the digital camera surface is [11; 15]  
\[ \begin{array}{ll}H(x,y)&=\left(O+R_{1}+R_{2}\right)\cdot\left(O+R_{1}+R_{2} \right)^{*}\\ &=\left|O\right|^{2}+\left|R_{1}\right|^{2}+\left|R_{2}\right|^{2}+OR_{1}^{*}+ OR_{2}^{*}+O^{*}R_{1}+O^{*}R_{2}\end{array}  \tag{1}\]  
In Equation (1), the first three terms correspond to the zero diffraction order, the fourth and fifth terms produce the virtual images, and the last two terms form the real images. Computing the Fourier transform of the acquired hologram, these terms appear spatially separated in the Fourier space, due to the off-axis configuration, as well shown in Figure 2b where the presence of some parasitic interferences are also highlighted [16; 29]. In order to recover the information of the real images, the corresponding spectra could be selected by two different spatial filters [15; 18]. As with classical holography, the reconstruction is obtained by multiply the digital hologram for a digitally computed reference wave and then the inverse Fourier transform of the spatial frequency components filtered is performed. By using the standard reconstruction algorithm on each filtered region, the two orthogonal components of the object beam (\(O_{x}\) and \(O_{y}\)), can be retrieved [15; 16; 18; 30]. So, the amplitude map and the phase map for each polarization component can be reconstructed. With the aim to evaluate the SoP change of the object beam due to the interaction with the sample under test, typically, two parameters are experimentally measured--the amplitude ratio \(\beta\), which is related to the different transmitted intensities of the two orthogonal components and corresponds to the azimuth of the polarization ellipse, and the phase difference \(\Delta\varphi\), that contains information on the different optical paths due to the refractive index anisotropy linked to the sample structure [18; 30]. So, the amplitude ratio angle can be evaluated as  
\[\beta\ =\ arctan\left(\frac{\left|O_{y}\right|}{\left|O_{x}\right|}\right) \tag{2}\]  
Equation (2) is obtained assuming that both the reference waves have the same intensity (\(\left|R_{1}\right|\) = \(\left|R_{2}\right|\)); this identity is experimentally achieved by controlling the orientation of the half-wave plates _HWP1_ and _HWP2_ in Figure 1[18; 30]. Regarding the phase difference between the orthogonal components of the object beam, it can be expressed as  
\[\Delta\varphi\ =\ phase\left(O_{y}\right)-phase\left(O_{x}\right)+\Delta \varphi_{R} \tag{3}\]  
where \(\Delta\varphi_{R}\) = \(phase\left(R_{2}\right)-phase\left(R_{1}\right)\) can be removed by a calibrated phase difference offset superimposed to the phase difference image [31]. The evaluation of \(\beta\) and \(\Delta\varphi\) allows the SoP of the sample under test to be univocally obtained; in other words, the distributions of the Jones vector at the surface of the sample under test can be retrieved from a single hologram acquisition [31]. For example, if \(\Delta\varphi\) = 0 or \(\pi\) a linear polarization is retrieved, while if \(\Delta\varphi\) = \(\pi\)/2 and \(\beta\) = \(\pi\)/4 or \(\Delta\varphi\) = \(-\pi\)/2 and \(\beta\) = \(\pi\)/4 a circular right or left polarization, respectively, are detected. Elliptical polarization states, i.e., intermediate polarization values, can be also measured.