## 2 Materials and Methods
### Analysis Methods
The numerical solution for the TIE in the case of a pure phase object (transparent object), is defined by Equation (7) [7], resulting from the standard equation of [PERSON] [27, 28]:  
\[\phi_{z}(x,y)=\mathbf{T}\mathbf{F}^{-1}\left[\frac{\mathbf{T}\mathbf{F}\!\left[-\frac{2\pi}{ \lambda}\frac{1}{I_{z}(x,y)}\frac{\partial I_{z}(x,y)}{\partial z}\right]}{4 \pi^{2}(u^{2}+v^{2})}\right] \tag{7}\]  
where \(\mathbf{T}\mathbf{F}^{-1}\) and \(\mathbf{T}\mathbf{F}\) denote the inverse Fourier Transform and Fourier Transform respectively, and \((u,v)\) denotes the spatial frequency variables corresponding to the coordinates \((x,y)\). Equation (7) is valid only for paraxial ray assumptions. The phase of the wave field is related to the variation of the image intensity along the optical axis through Equation (7). In practice, the intensity derivative \(\partial I_{z}(x,y)/\partial z\) in the observation plane \(z\) cannot be measured directly; we adopted the following approximation (Equation (8)) using two defocused images, one above the observation plane (\(I(x,y,z+\Delta z)\)) and one below (\(I(x,y,z-\Delta z)\)):  
\[\frac{\partial I_{z}(x,y)}{\partial z}\approx\frac{I(x,y,z+\Delta z)-I(x,y,z- \Delta z)}{2\!\cdot\!\Delta z} \tag{8}\]  
where \(\Delta z\) is the defocus distance. Phase distortions induced in the wave field as the wave propagates from point \(r_{z_{0}}(x,y)\) to \(r_{z}(x,y)\) can be related to the refractive index of the specimen through solutions to the eikonal equation [8, 29] for the phase:  
\[\phi_{z}(x,y)-\phi_{z_{0}}(x,y)=\frac{2\pi}{\lambda}\int_{z_{0}}^{z}\Delta n_ {z^{\prime}}(x,\ y)dz^{\prime} \tag{9}\]  
\(\Delta n_{z}(x,\ y)\) is the refractive index difference between the specimen and its surrounding medium (specimen layer). Taking the specimen to be located directly above the origin \(r_{z_{0}}(x,y)\), hence \(\phi_{z_{0}}(x,y)=0\), and for points inside the specimen, it is found that [8]:  
\[\phi_{z}(x,y)=\frac{2\pi}{\lambda}\int_{0}^{z}\Delta n_{z^{\prime}}(x,\ y)dz^{\prime} \tag{10}\]  
Therefore, if \(\phi_{z}(x,y)\) is known in multiples planes (with small value of defocus distance \(\Delta z\)) inside the specimen, its 3D refractive index distribution can be recovered via Equation (11) [30] through sequential method (using finite differences) applied to pairs of consecutive planes (one pair after one sequentially) with the same value of defocus distance:  
\[n_{z}(x,y)=n_{m}+\frac{\lambda}{2\pi}\ \frac{\partial\phi_{z}(x,y)}{\partial z} \tag{11}\]  
where \(n_{m}\) is the refractive index of the specimen layer. To retrieve the 2D phase \(\phi_{z}(x,y)\) corresponding to the observation plane \(z\), we used Equation (7) and three recorded images of the sample were needed, one in the observation plane, and the both of the others below and above the observation plane with a defocus distance \(\Delta z\). This procedure was repeated to obtain \(\phi_{z}(x,y)\) in multiples planes by changing the observation plane with the same value of \(\Delta z\) until the whole specimen was scanned along the optical axis. The defocus distance is an important factor in the quality of the reconstruction. When \(\Delta z\) was large, the hypothesis of linearity that enables the above approximation (Equation (8)) of the intensity derivative \(\partial I_{z}(x,y)/\partial z\) was not valid, and this nonlinearity affected the accuracy of the reconstruction. For small values of \(\Delta z\) where the linearity was preserved, there was a reduction of signal to noise ratio due to the term \(I(x,y,z+\Delta z)-I(x,y,z-\Delta z)\), which becomes smaller. This required a compromise between the defocusing, accuracy, and signal to noise ratio. Therefore, when we are in the presence of multiple plane images and we would like to retrieve the 2D phase \(\phi(x,y)\) corresponding to the focal plane, we cannot simply use pairs of plane images to recover the phase from average of them [13]. In this work, we used the method proposed by [PERSON] et al.