## 2 Theory
### The Magnifications \(M_{L}\) and \(M_{T}\) of the Images in the In-Line Interferometer
#### 2.1.1 Registration of the Superimposed Wavefronts  
The process of recovering the focused image, which is conducted in two stages, is described as follows. The first stage is the filtering of the spatial-frequency spectrum with the DSB filter, the magnification of the filtered image, and the intensity registration. The second stage involves digital processing of the intensities to retrieve the complex amplitude of the diffracted image and numerically propagating it to the reconstruction plane to obtain the magnified focused image. The diffracted wavefront propagates from the plane \((x_{0},y_{0})\) to the plane \((x_{1},y_{1})\), at a distance \(d~{}=~{}\Delta z_{o}\). This propagation can be described by using the Fresnel diffraction equation [23; 24; 25]:  
\[u_{o}(x_{1},y_{1})~{}=~{}\frac{\exp(ik\Delta z_{o})}{i\lambda\Delta z_{o}} \iint\limits_{-\infty}^{\infty}u(x_{0},y_{0})\exp\biggl{\{}i\frac{k}{2\Delta z_ {o}}\Bigl{[}(x_{0}-x_{1})^{2}+(y_{0}-y_{1})^{2}\Bigr{]}\biggr{\}}dx_{0}dy_{0}. \tag{10}\]  
where \(k~{}=~{}2\pi/\lambda\) is the wave number. The superposition between the diffracted and collimated wavefronts in the plane \((x_{1},y_{1})\), as is shown in Figure 2, can be written as:  
\[u_{o}(x_{1},y_{1})~{}=~{}1+u_{o}(x_{1},y_{1}). \tag{11}\]  
Note that the assumption given in Equation (11) is a good approach when the analyzed objects are small in comparison with the size of the illuminating beam, as is the case of experiments provided in further sections, but will not be a suitable approach in other scenarios, as for instance, for quantitative phase imaging. The frequency spectrum of the superposition of wavefronts is observable in the focal plane of the lens \((\eta,\xi)\) and can be mathematically represented as the Fourier transform of Equation (11):  
\[Ut_{o}(\eta,\xi)~{}=~{}\delta(\eta,\xi)+U_{o}(\eta,\xi). \tag{12}\]  
At the focal length of the lens, the double-sideband filter blocks the first half of the frequency spectrum corresponding to \(\eta>0\). The superposition of the wavefronts, with half of the frequency spectrum blocked, can be computed as the inverse Fourier transform of Equation (12) with the integration boundaries in the \(\eta\) direction ranging from \(-\infty\) to \(0\). The resulting image \(u_{i}(x_{2},y_{2})\) is a scaled and inverted version of \(u_{o}(x_{1},y_{1})\):  
\[u_{i}(x_{2},y_{2})_{\eta>0}~{}=~{}\exp[ik(s_{o}+s_{i})]\frac{1}{\mathrm{M}_{ \mathrm{T}}}u_{o}\biggl{(}\frac{x_{2}}{\mathrm{M}_{\mathrm{T}}},\frac{y_{2}}{ \mathrm{M}_{\mathrm{T}}}\biggr{)}_{\eta>0}. \tag{13}\]  
Here, \(u_{o}(x_{2}/\mathrm{M}_{\mathrm{T}},y_{2}/\mathrm{M}_{\mathrm{T}})\) is the amplified and inverted ideal image of the diffracted wavefront \(u_{o}(x_{1},y_{1})\) under the assumption that lens aberrations are neglected and the diameter of the lens pupil is larger than the transverse section of the wavefront. The recorded intensity in the plane \((x_{2},y_{2})\) corresponds to the squared modulus of \(u_{i}(x_{2},y_{2})\):  
\[|u_{i}(x_{2},y_{2})|^{2}_{\eta>0}~{}=~{}\frac{1}{\mathrm{M}_{\mathrm{T}}^{2}} \biggl{|}u_{o}\biggl{(}\frac{x_{2}}{\mathrm{M}_{\mathrm{T}}},\frac{y_{2}}{ \mathrm{M}_{\mathrm{T}}}\biggr{)}_{\eta>0}\biggr{|}^{2}. \tag{14}\]  
Figure 2: Scheme of the recording stage of the overlap between the diffracted and illumination wavefronts. Analogously, the other half of the frequency spectrum, corresponding to \(\eta<0\), is blocked at the Fourier plane to obtain the subsequent intensity:  
\[|u_{i}(x_{2},y_{2})|_{\eta<0}^{2} = \frac{1}{\text{M}_{\text{T}}^{2}}\Bigg{|}u_{o}\bigg{(}\frac{x_{2}} {\text{M}_{\text{T}}},\frac{y_{2}}{\text{M}_{\text{T}}}\bigg{)}_{\eta<0}\Bigg{|} ^{2}. \tag{15}\]  
Finally, both intensities given by Equations (14) and (15) are combined to obtain the intensity corresponding to the object's full spectrum.