## 2 Theory
### The Magnifications \(M_{L}\) and \(M_{T}\) of the Images in the In-Line Interferometer  
In an in-line interferometer, a microscopic object is illuminated with a collimated wavefront. Both the wavefront diffracted by the object and the wavefront collimated follow the same paths and overlap with each other. In accordance with Figure 1a, the microscopic object is located at the plane \((x_{0},y_{0})\), and the superposition between the diffracted and collimated wavefronts take place at a distance \(\Delta z_{o}\) in the plane \((x_{1},y_{1})\). The lens L forms the magnified image of the wavefront superposition at the plane \((x_{2},y_{2})\), where a recording medium, from now on a CCD camera, is positioned. The transverse magnification, \(\rm M_{T}\), and the longitudinal magnification, \(\rm M_{L}\), associated with the planes \((x_{1},y_{1})\) and \((x_{2},y_{2})\), respectively, are calculated as:  
\[\rm M_{T}\ =\ \frac{y_{2}}{y_{1}}\ =\ \frac{x_{2}}{x_{1}}\ =\ -\frac{s_{i}-f}{f}, \tag{1}\]  
\[\rm M_{L}\ =\ \frac{dz_{i}}{dz_{o}}\ =\ -\frac{(s_{i}-f)^{2}}{f^{2}}\ =\ -(\rm M_{T})^{2}. \tag{2}\]  
The Equations (1) and (2) express the transverse and longitudinal magnifications of the images, respectively, as a function of the focal length \(f\) and the distance \(s_{i}\) and establish a relation between \(\rm M_{T}\) and \(\rm M_{L}\). Note that if the distance between the lens L and the CCD camera placed at the plane \((x_{2},y_{2})\) remains unaltered, both magnifications remain unchanged. The spatial-frequency spectrum of the superimposed wavefronts is observable at the focal length of the lens in the Fourier plane \((\eta,\xi)\). Thus, the double-sideband filter (DSB) is positioned in such a Fourier plane. The DSB filter first blocks one half of the frequency spectrum and, subsequently, the other half. Consequently, each of the two images formed by the lens in the plane \((x_{2},y_{2})\) possesses half of the frequency spectrum. The experimental scheme used to implement the DSB is shown in Figure 1b. The collimated wavefront is linearly polarized at \(45^{\circ}\) by using the linear polarizer LP1. In front of the CCD, another linear polarizer LP2 is positioned, aligning its transmission axis at the same angle. The spatial light modulator SLM-LC is situated in the focal plane of the lens L to block the half of the frequency spectrum in the following manner: a binary distribution of grey-level values is addressed to the SLM-LC in such form that half of the SLM-LC screen induces a phase retardance of \(\delta 1\ =\ 0^{\circ}\), while the other half induces a phase retardance of \(\delta 2\ =\ 180^{\circ}\), as shown in the part (I) of Figure 1b. In this way, the polarization state of the linearly polarized beam at \(45^{\circ}\) remains unchanged when passing through the half of the SLM-LC with zero retardance. Consequently, the beam is transmitted without any issues by linear  
Figure 1: (**a**) Scheme of the in-line interferometer and (**b**) sketch of the double-sideband filter (DSB). polarizer LP2. However, the beam that passes through the half of the SLM-LC screen with a retardance of 180\({}^{\circ}\) undergoes a rotation of 90\({}^{\circ}\) in its polarization state, resulting in linear polarization at 135\({}^{\circ}\). As a result, this beam is blocked by the polarizer LP2. The frequency spectrum is centered at the boundary between the phase retardances \(\delta\)1 and \(\delta\)2 on the SLM-LC, which creates a knife-edge effect, blocking half of the frequency spectrum. Subsequently, a similar procedure is followed, where the phase retardances are interchanged to \(\delta 1\ =\ 180^{\circ}\) and \(\delta 2\ =\ 0^{\circ}\) and addressed to the SLM-LC, as shown in part (II) of Figure 1b, to block the other half of the frequency spectrum. More details about the DSB polarization-based filter implementation can be consulted in Refs. [20; 21].