## 2 Theoretical Background
### PS-DHI for Stokes Vectors or Miller Formalism
Basically, two classes of experimental setup based on the geometric phase can be found in the recent literature to evaluate the full SoP of an optical beam. In the first, shown in Figure 5a, a triangular common-path interferometer (TCPI) is used to generate two interferograms which are aligned together on a single charge coupled device (CCD) target [40]. This is made possible by dividing the object beam into two orthogonal circularly polarized components (left-handed and right-handed) through the TCPI, and then these two components interfere with a reference beam. However, this implementation is difficult to align, leading to low measurement resolution, reduced field of view (due to separated fringe pattern for the orthogonal components recorded in two different region of the same CCD), and image matching errors. The second setup, reported in Figure 5b, is based on a hybridpolarization-angular multiplexing digital holographic approach (PAMDH) [47], implemented by a double-channels Mach-Zehnder interferometer. In this case, two orthogonal and linearly polarized reference beams interfere with the object beam. In both cases, the acquisition of a composite hologram generated by combining two patterns of interference fringes with distinct orientations allows the contemporary estimation of the orthogonal polarization components for an optical wave. Once these components are retrieved, the Stokes parameters can be evaluated by applying the geometric phase theory [47]. The main difference between two reported schemes is that in the setup depicted in Figure 5b the multiplexed hologram covers the whole area of the CCD, while in the solution presented in Figure 5a, a smaller field of view is achieved to avoid the influence of the change in the intensity and polarization distribution of the reference wave. For this configuration, the field of view is determined by the region of two images on CCD related to the two orthogonal components of the object wave separated at a distance controlled by the TCPI  
Figure 5: Experimental setup for polarization measurement—(**a**) triangular common-path interferometer (TCPI). GTP: Glan–Taylor polarizer; BS1 and BS2: beam splitter; PCS: polarization conversion system; \(\lambda\)/4: quarter-wave plate; PBS: polarizing beam splitter; M: mirror; RT: reversed telescope; CCD: charge-coupled device. The elements inside the dashed box form a TCPI (Reproduced with permission from [PERSON] et al. [40], AIP, 2019); (**b**) schematic of the PAMDH system based on the geometric phase. BE: beam expander; L: lens; P: polarizer; S: sample; HWP: half-wave plate; NPBS\({}_{1}\)-NPBS\({}_{2}\): non-polarized beam splitters; PBS\({}_{1}\)-PBS\({}_{2}\): polarized beam splitters; M\({}_{1}\)-M\({}_{3}\): mirrors; NDF: neutral density filter; MO: microscope objective; TL: tube lens (reprinted from [PERSON] et al. [47]). Therefore, since the arrangement of Figure 5b is more useful to produce and control a larger field of view, for simplicity of discussion, here, the basic theory related to this scheme is reported. In the PAMDH approach, Equation (1) becomes [47]  
\[ \begin{array}{ll}H(x,y)&=(O+R_{V}+R_{H})\cdot(O+R_{V}+R_{H})^{*}\\ &=|O_{V}|^{2}+|O_{H}|^{2}+|R_{V}|^{2}+|R_{H}|^{2}+O_{V}R_{V}^{*}+O_{H}R_{H}^{*} +O_{V}^{*}R_{V}+O_{H}^{*}R_{H}\\ &=I_{V}+I_{H}\\ \end{array}  \tag{5}\]  
\[ \begin{array}{ll}I_{V}&=&|O_{V}|^{2}+|R_{V}|^{2}+2|O_{V}||R_{V}|\cos\varphi_ {V}\\ I_{H}&=&|O_{H}|^{2}+|R_{H}|^{2}+2|O_{H}||R_{H}|\cos\varphi_{H}\\ \end{array}  \tag{6}\]  
where \(\varphi_{V}\) and \(\varphi_{H}\) are the phase variations related to each of the two orthogonal components of the reference beams and object waves. Therefore, the two orthogonal complex amplitudes \(O_{V}\) and \(O_{H}\) of the object wave can be numerically retrieved. To reconstruct the SoP, the polarization state should be reported on the Poincare sphere.