## 3 Results and Discussion  
During the computation of the PSF for our experimental device, two parameters were not directly accessible: the depth \(z_{p}\) of the point object in the specimen layer, and the refractive index \(n_{c}\) of the immersion medium (air) during the experimentation [25]. Our sample of microspheres of diameter 1 \(\upmu\)m was allowed to dry during experimentation. This enabled us to assume that the point object was on the interface of air and specimen layer, so \(z_{p}=0\). The measured pressure and temperature in the measurement environment (\(P=992\) hPa, \(T=17\)\({}^{\circ}\)C) enabled the determination of the refractive index, \(n_{c}\), of the immersion medium (air) during experimentation; \(n_{c}=1.00024\) by using Equation (13) [33,34]:  
\[n_{c}=1+\left(6.4328\cdot 10^{-5}+\frac{2.94981\cdot 10^{-2}}{146-\lambda^{-2}}+ \frac{2.554\cdot 10^{-4}}{41-\lambda^{-2}}\right)\frac{15\cdot P}{T} \tag{13}\]  
where \(P\), \(T\), and \(\lambda\) are in bar, in degrees Celsius and in \(\upmu\)m respectively. Figure 2 presents the PSF of our device. The images of the sample in different planes were affected by the contribution of out-of-focus planes (Figure 3a-c). Restoration of these images was achieved by using the Richardson-Lucy algorithm [35,36] implemented in MatLab, taking into account the 3D-PSF. The number of iterations used was 25. This led to a reduction not only in the effects of diffraction but also in the noise and pixel intensity, due to the contribution of the adjacent planes (Figure 3d-f). Applying the TIE method sequentially on the corrected images led to an extracted refractive index of the specimen (Figure 4a-g). We can observe in Figure 4a-g that the outlines of the microsphere became smaller as we moved away from the focal plane (\(z=0\)\(\upmu\)m). This was due to effect of sectioning. In a plane of the microsphere, the spatial distribution of its refractive index was not homogeneous. The homogeneity of refractive index distribution was drastically damaged as we moved away from the focal plane. This was due to diffraction effects [37] that became prominent. In this case, the paraxial ray assumptions, which were required to apply TIE, were nullified [8]. This induced artefacts in the refractive index reconstruction (Figure 4a-c). TIE method drastically failed to recover the 3D refractive index distribution of the sample for an optical system with a high numeric aperture, as was the case for our device (\(NA_{ob}=0.75\)) [37]. Figure 3: Recorded images in the different planes of a polystyrene microsphere of diameter 1 \(\upmu\)m at the wavelength \(\lambda=640\) nm (**a**–**c**), and the corresponding restored images (**d**–**f**). Figure 2: X–Z cross section of the computed point spread function (PSF) with the acquisition parameters of the experimental device: the numeric aperture of the dry objective \(NA_{ab}=0.75\) with working \(WD=0.71\) mm; the refractive index of specimen layer \(n_{m}=1.33\); the standard value of refractive index of the air \(n_{c^{\ast}}=1\); the refractive index of the air during the experimentation \(n_{c}=1.00024\), \(z_{p}=0\); the pixel size in the object space is 0.1 \(\upmu\)m.  
A reconstruction of the refractive index after the deconvolution of the corrected images by the use of the 3D-PSF is presented in Figure 4h-n. The sectioning effect was more greatly appreciated through the spatial distribution of refractive index. The circular outlines of the microsphere were well highlighted. The artefacts that were observed in the planes \(z=-0.4\)\(\upmu\)m, \(z=-0.3\)\(\upmu\)m, \(z=-0.2\)\(\upmu\)m were removed. In each plane of the microsphere, the homogeneity of refractive index distribution was greatly improved and tended to cover all of the circular outline of the microsphere, mostly in the focal plane. Table 1 presented a comparison between the mean values of the microsphere's refractive index in the presence of the diffraction, and in the absence of diffraction.