## 2 Theoretical Background
### PS-DHI for Jones Formalism  
In most polarimetric techniques, SoP parameters can be evaluated by applying more or less complex algorithms to various images acquired with different settings of polarization-analyzing components (polarizers, rotators, and retarders). Since these procedures need several rotations of the analyzing optics, the acquisition time is very long compared to the performances of a digital camera. Several solutions were proposed in the literature to improve the temporal resolution, such as the possibility to use a liquid-crystal universal compensator [23]; however, the goal was reached only with techniques that allow recording all parameters of the polarization state through the acquisition of a single image. Hence, since the pioneering paper published by [PERSON] and [PERSON], which generated the interference between two orthogonal linearly polarized reference waves and an object wave [26] using a Mach-Zender interferometer, several other published works followed this approach [15; 27; 28]. The typical experimental configuration for the recording of polarization holograms is illustrated in Figure 1. It consists of a modified Mach-Zehnder interferometer with two reference waves--\(R_{1}\) and \(R_{2}\)--that interfere with the object beam \(O\)[11; 15; 16; 17]. Two operating conditions are possible [18]. First, the object plane beam is linearly polarized by a polarizer (\(P1\) oriented at \(45^{\circ}\)) and a quarter wave plate (\(QWP1\) oriented at \(0^{\circ}\) respect the incoming light). Due to the passage through the sample, the state of polarization of the beam \(O\) can change. An objective lens is used to collect the wave emitted from the sample. A polarized beam splitter (\(PBS\)) allows the user to obtain two orthogonal linearly polarized reference beams \(R_{1}\) and \(R_{2}\), where orthogonality avoids any interference between the reference beams. Additionally a couple of polarizers or quarter waves plates (\(QWP2\) and \(QWP3\)) preserve the linear polarization when their fast axes are aligned parallel respect to the polarization states of the respective reference waves. In the second operating condition, a circularly polarized light is incident on the sample by orienting the \(QWP1\) fast axis angle at \(-45^{\circ}\) (left-handed circular polarization) or \(+45^{\circ}\) (right-handed circular polarization) respect to the polarizer transmission axis \(P1\). The two orthogonal linearly polarized reference are transformed in right and left circularly polarized beams by the quarter wave plates \(QWP2\) and \(QWP3\) oriented of \(+45^{\circ}\) and \(-45^{\circ}\) respect to the polarization states of the two references waves, respectively. The beam splitter \(BS1\) allows to overlap \(O\), \(R_{1}\), and \(R_{2}\) beams, and the interference among these waves gives rise to the hologram that is acquired by a digital camera in an off-axis configuration, i.e., with the three waves propagating along slightly different directions (as highlighted in the inset in Figure 1). The intensities of \(O\), \(R_{1}\) and \(R_{2}\) can be adjusted by the half-waves plates \(HWP1\) and \(HWP2\), while the angles of incidence of \(R_{1}\) and \(R_{2}\) can be controlled by mirrors \(M2\) and \(M3\), respectively. An example of the recorded digital hologram in shown in Figure 2a. In the insert, where a magnification of the interference between object and the two orthogonal reference waves is reported, the two sets of fringe patterns are clearly visible. Figure 1: Basic scheme of the polarization-sensitive digital holographic imaging (PS-DHI) experimental setup. Abbreviations: M: mirrors; BS: beam splitter; PBS: polarized beam splitter; P: polarizer; QWP: quarter-wave plate; HWP: half-wave plate. In the inset, the incident directions and the polarization states of the object and reference waves are highlighted.