## 3 PS-DHI Applications
### Biological
It is evident that \(\Delta\varphi\) map shows a birefringence distribution (bright pattern) over all the sperm head in the control sperm cell, while the \(\Delta\varphi\) map for reacted sperm cell presents a reduction of the birefringent distribution, that is confined in the post-acrosomal area. The acrosome reaction was induced by a heparin treatment. A statistical analysis of the distribution of birefringence patterns of sperm from three donors exposed to heparin for 0 h (control sample) and 4 h (reacted sample) was performed, and the results are resumed in Figure 8c. By combining PS-DHI and RS, the authors proposed a new fully label-free protocol for the recognition of healthy and reacted sperm cells. In detail, sperm cells with a head entirely birefringent are selected by PS-DHI, assuring their integrity; then, the heparin treatment was performed on these chosen spermatozoa to induce the acrosome reaction. Finally, the effectively reacted spermatozoa are selected by estimating again their polarization state by the PS-DHI combined with the study of their Raman spectra. This interesting combined approach leads to the identification of spermatozoa in which the modification in theirbirefringence distribution is imputed only to the acrosome reaction, while those in which this variation is correlated with defects are not considered [30]. In 2019, [PERSON] et al. [73] proposed a proof of concept of a holographic fiberscope that allows producing full-field images of amplitude and quantitative phase in two polarizations, using a novel parallelized transmission matrix approach. The polarimetric imaging of birefringent and deattenuating samples was carried out to verify the feasibility of this approach. Due to their small diameter and high flexibility, imaging through optical fibers is already implemented in biomedical endoscopy and industrial inspection. Therefore, the introduction of a holographic fiberscope for birefringence measures appears very interesting for biological applications and remote sensing. Figure 8: (**a**) Amplitude parameter \(\beta\) and phase difference \(\Delta\varphi\) relative to a control sperm cell (0 h in heparin). Colorbars indicate the mapping of the phase variation. (**b**) The same polarization parameters measured for a reacted sample (4 h in heparin). (**c**) Distribution of birefringence patterns of sperm from three donors exposed to heparin for 0 h (control sample) and 4 h (reacted sample). Scale bar: 4 \(\upmu\)m (adapted from [PERSON] at al. [30]). An application worthy of note is that studied by [PERSON] et al. [74; 75]. They developed a dual-view polarization-resolved digital holographic systems for particle tracking [74] as well as for define if a particle has a spherical shape or not and to estimate its size [75]. This novel system allows to image the same volume from two perpendicular directions, giving information about the amplitude ratio angle \(\beta\) from two views. The authors found that the size of a non-spherical particle can be estimated from \(\beta\) with an upper limit of about nm. This approach could be very useful to detect and distinguish different particles, including biological particles, under different flow conditions by estimating their polarization response. Finally, the instantaneous (single-shot) measure of the spatial variations of the phase retardance induced in either geometric or dynamic phase is carried out through an alternative approach presented in Ref [76], where a quantitative fourth-generation optics microscopy (Q4 GOM) has been developed; even though this approach doesn't characterize the full SoP, thanks to its unique optical performance, it can open new research in diagnosis of optical composite nanostructures or biomolecular sensing. The phase restoration is based on the self-interference of optical wave and is achieved in an intrinsically stable common-path setup. Basically, in the live cell imaging an add-on fourth-generation (4G) optics imaging module is combined to a polarization adapted interference microscope, as shown in Figure 9a.