## 3 PS-DHI Applications
### Biological  
The order of molecular architecture can play a role in the dependence of the refractive index on the polarization and propagation direction of the optical beam. Indeed, the anisotropy of the refractive index, i.e., birefringence, could be related to the presence of filament arrays and/or membranes (made of a lipid bilayer that exhibits some degree of orientation) included in organelles and cells. Several pathological modifications, due, for example, to physical damage or disease such as cancer, may modify optical properties of biological tissues by altering their structure and, thus, leading to a change in their birefringence pattern [54; 55; 56]. Therefore, the detection of these modifications could become a valuable tool to identify the molecular order, follow events, and diagnose diseases. Typically, the characterization of birefringence is carried out by using quantitative polarized light microscopy [57; 58; 59], polarimetry [60; 61; 62; 63], and experimental determination of the Mueller matrix [64; 65; 66]. Since these techniques require the acquisition of several images to retrieve the birefringence of the sample, they appear too slow for living semi-transparent biological sample imaging. For this reason, PS-DHI may have a good chance of being used as a label-free and fast technique (only one acquired image) for the study of the SoP of biological samples. However, this technique has started to be used for biological imaging only in the last few years; therefore, the scientific papers on this field of application present in the literature are still a low number. For example, [PERSON] et al. [67] studied the birefringence distribution of biological tissues by using polarization-dependent phase-shifted holograms. Even if the authors used a setup based on a modified Mach-Zehnder interferometer, which allows the recording polarization holograms by rotating a polarizer and so requiring multiple acquisitions, this approach takes advantage by DH and, thus, can be exploited to have only one image acquisition. Interestingly, the results demonstrated that the median birefringence value of cancerous bladder tissues is higher than that of the normal bladder tissues. Hence, this approach can be effectively used to discriminate between cancerous and non-cancerous tissues. PS-DHI in a configuration similar to that shown in Figure 1 has also been used to distinguish among three different B-leukemia transformed cell lines, providing a diagnosis method of acute lymphoblastic leukemia type B, a cancer with a high mortality rate that affects B lymphocytes [68]. The same approach has been applied also to human sperm cells [30]. In fact, in normal morphological human sperm cells heads, due to longitudinally oriented protein filaments, there is a strong birefringence [69]. When the acrosome reaction occurs, i.e., spermatozoa are ready to approach the egg, the local protein organization disaggregates and, as a consequence, there is a variation in the intrinsic birefringence properties. In fact, reacted spermatozoa show a partial head birefringence, typically in the post-acrosomal region [70]. In the paper published by [PERSON] et al. [30], PS-DHI is proposed in a configuration combined with Raman spectroscopy (RS) for a complete, accurate and label-free estimation of the biological proprieties of fixed air-dried sperm. Indeed, PS-DHI provides quantitative information on the cell morphology, motility and SoP [8; 71], whereas the RS technique gives complementary specific biochemical fingerprint of the sample, without harming the integrity of live specimens [72]. In Figure 8a,b, the amplitude parameter \(\beta\) and phase difference \(\Delta\varphi\) relative to a control sperm cell and to reacted sample retrieved by PS-DHI are reported.