## 2 Theoretical Background
### PS-DHI for Stokes Vectors or Miller Formalism  
Jones vectors cannot be used to describe light that does not remain in a single polarization state. So, to treat fully, partially, or unpolarized light, Stokes parameters (S\({}_{1}\),S\({}_{2}\),S\({}_{3}\)) are often used. In fact, unlike the Jones vectors, the Stokes parameters can define all kinds of SoPs of the optical beam. Unfortunately, as in the case of Jones formalism, the typical methods for estimating the Stokes parameters of an optical wave require to register several intensity distributions at different detection states via rotating the polarized optical elements. Generally, these methods require time-sequential operations using an arrangement with a rotating waveplate and a fixed analyzer. However, system error could be generated by the rotating elements, due to their inhomogeneous transmittance, and obviously, fast acquisitions are not possible [34; 35]. Other proposed methods need multichannel simultaneous measurement; in this case, the amplitude or the wavefront is split into several channels, each one is analyzed by employing appropriate polarization optical components thus leading to a complicated and cumbersome dynamic measurement system [36; 37]. Moreover, systematic and calculation errors are induced by a not exactly precise image matching. Thus, it is clear that the development of imaging polarimeters in real-time without the demand for mechanical or active elements for polarization control is still a valuable aim. With this purpose, several approaches were proposed; in one of these, multiple interference patterns are produced at the surface of a digital camera and the information of a different polarized component of the optical beam under investigation is linked to fringes patterns having different spatial frequencies. The Stokes parameters can be estimated through demodulation of the obtained image by a Fourier transform approach [38]. Recently, the combination of DH with theory of the Pancharatnam-Berry (PB) phase has paved the way toward the implementation of a new method for evaluating the state of polarization of arbitrary waves with a single exposure of the interference pattern and a quick acquisition for one object wave with no moving optical components [39; 40]. [PERSON] [41] and [PERSON] [42] introduced the so-called PB phase of an optical wave that is taken along a closed cycle on the Poincare sphere. With this formalism, polarization transformations give rise to two optical phase retardation--one related to optical path difference (called dynamic phase) and an extra one which is equal to minus half of the solid angle subtended by the closed path on the sphere. Therefore, this extra phase, i.e., the PB phase, depending only on the geometry of the transformations' path of on the Poincare sphere, is also called \"geometric phase.\" Nevertheless, the PB phase occurs generally in the following two conditions: when there is a variation of polarization state in the beam propagation, and when there is a variation of the mode structure of the beam propagation [43; 44]. Regarding the first condition, the PB phase and the polarization state variation are quantitatively related, giving the possibility to estimate the SoP from the PB phase measurement. Even though DH allows quantitative phase retrieve of the object beam to be performed [45; 46], when the phase difference contains both the PB phase (related to the SoP) and dynamic phase (related to the optical density of the sample), these two phases are indistinguishable for the reconstruction process. To separate the two contributions, two holograms should be generated--one to retrieve the dynamic phase and the other one for the geometric phase, respectively. The dynamic phase is then used for the evaluation of the refractive index or 3D shape of the sample under investigation, whereas its interaction with the polarized light is estimated through the geometric phase.