## 2 Theoretical Background
### PS-DHI for Jones Formalism
Considering the polarization ellipse represented in Figure 3 and that corresponds to the projection of the trajectory of the extremity of the vector \(O\) on the plane xy, it can be characterized by the parameters \(\psi\) (orientation angle) and \(\chi\) (ellipticity angle), which can be additionally evaluated by the following equations [15; 17]:  
\[ \begin{array}{ll}\psi\ =\ \frac{1}{2}arctan\left(\frac{2\left|O_{x} \right|\left|O_{y}\right|\cos\left(\Delta\varphi\right)}{\left|O_{x}\right|^{2 }-\left|O_{y}\right|^{2}}\right)\ =\ \frac{1}{2}arctan\left[\tan\left(2\beta\right)\cos\left( \Delta\varphi\right)\right]\\ \chi\ =\ \frac{1}{2}arcsin\left(\frac{2\left|O_{x}\right|\left|O_{y}\right|\sin \left(\Delta\varphi\right)}{\left|O_{x}\right|^{2}+\left|O_{y}\right|^{2}} \right)\ =\ \frac{1}{2}arcsin\left[\sin\left(2\beta\right)\sin\left( \Delta\varphi\right)\right]\end{array}  \tag{4}\]Regarding the direction of rotation of the vector \(O\), it is defined as left-hand polarization (L-state) or right-hand polarization (R-state) depending on whether \(-\pi\leq\Delta\varphi\leq 0\) or \(0\leq\Delta\varphi\leq\pi\), respectively. Finally, in a recent and very interestingly work a new compact experimental setup has been proposed as proof of concept [32]. The birefringence distribution was measured by the interference fringes based on three circularly polarized beams--two mutually orthogonal polarized (left-handed and right-handed) reference waves and a right-handed object wave. These three-beam interfering fields were obtained by monolithic gratings, so all waves were crossed at the same angle on the hologram plane, generating two sets of the fringe pattern. With this approach, all the optical elements required to obtain the three beams in the set-up illustrated in Figure 1 can be replaced by three grating vectors; moreover, in order to have practical interferometry, a monolithic grating containing three diffractive gratings positioned in a threefold-symmetric arrangement has been employed [32,33]. A schematization of both the monolithic gratings and the operating principle is illustrated in Figure 4. When a plane beam impinges on the monolithic gratings, the first-order diffractions of each grating intersect the hologram plane with the same angles. As described, analyzing the interference patterns allows the estimation of the anisotropic phase shift in the object beam. Thus, this arrangement has the potentiality to evaluate a two-dimensional birefringence distribution in a single shot and compact way. Figure 4: Representation of (**a**) the monolithic gratings, (**b**) interference of the diffracted beams, (**c**) monolithic gratings with the half-wave plate (HWP) and sample of birefringent medium, and (**d**) experimental setup designed for measuring birefringence distributions (Reproduced with permission from [PERSON] et al. [32], © The Optical Society, 2018). Figure 3: Polarization ellipse. The ellipticity is given by the ratio of the length of the semiminor axis to the length of the semimajor axis, b/a = tan(\(x\)). The ellipse is further described by its azimuth \(\psi\), measured counterclockwise from the x axis [10].