## 3 Simulation of Two Objects in Different Longitudinal Planes: Mixing of Opaque and Phase Objects  
In this section, two irregular objects are generated in two different longitudinal planes. Each object has its own size and shape as shown in Figure 2. The object in plane 1 is a phase object (left of Figure 2) and the object in plane 2 is an opaque object (right of Figure 2). The numerical setup is presented in Figure 3. We evaluate separately the two diffraction patterns created by object 1 and by object 2 respectively. As both grids do not necessarily have the same pixel size (due to sampling conditions used when realizing both numerical Fresnel integrals), we use an interpolation function to super-impose both diffraction patterns. Noise can be added as well. As in [20], we assume that multiple scattering can be neglected (_i.e._, the part of the light diffracted by the first object and then by the second object). The incident beam is a Gaussian beam. The initial beam waist is \(\omega_{0}=2.3\) um. The wavelength is \(\lambda=642\) nm. The simulated configuration respects \(e_{1}=58.8\) mm, \(f_{1}=42.8\) mm, \(e_{2}=342.84\) mm, \(f_{2}=5.5\) mm. The first object is located at distance \(\triangle_{1}=9.5\) mm and the second object is located at distance \(\triangle_{2}=10\) mm from the second lens as described in Figure 3. The distance between the second lens and the CCD sensor equals \(18\) mm. For the two objects, the system introduces magnification factors between the object plane and the CCD plane. They are given analytically by the following relation:  
\[g_{1,2_{x,y}}=\frac{w_{lot_{x,y}}}{w_{1,2_{x,y}}} \tag{11}\]  
where \(g_{1,2_{x,y}}\) are the magnification factors between Plane 1 (or Plane 2) and the CCD plane. \(w_{1,2_{x,y}}\) are the beam waist dimensions (along x and y, respectively) of the unperturbed Gaussian beam in Plane 1 (or in Plane 2 respectively). \(w_{tot_{x,y}}\) are the beam waist dimensions of the unperturbed Gaussian beam in the plane of the CCD sensor. They are given by:  
Figure 3: Simulated set-up. Figure 2: Simulated phase particle (**left**) with phase shift \(\varphi=0.3\)\(\pi\) and opaque particle (**right**). \[w_{tot_{x,y}}=\left(\frac{\lambda B_{tot}^{x,y}}{\pi\omega_{0}}\right)\left[1+ \left(A_{tot}^{x,y}\frac{\pi\omega_{0}^{2}}{\lambda B_{tot}^{x,y}}\right)^{2} \right]^{1/2} \tag{12}\]  
\(A_{tot}^{p}\) and \(B_{tot}^{p}\) (with \(p=x\) or \(y\)) are the matrix elements of the transfer matrix describing the whole system between the laser source and the CCD sensor: \(M_{tot}^{p}=M_{2}^{p}\times M_{1}^{p}\). In the simulation, the sensor size is \(1024\times 1024\) pixels and each pixel is \(4.4\times 4.4\)\(\upmu\)m\({}^{2}\). The magnification factor introduced by the system for the object in Plane 1 is equal to \(6.8\), while it is equal to \(5.3\) for the object in plane 2. Figure 4a is the hologram of the two objects recorded by the CCD sensor. Figure 4b,c show the optimal reconstruction of the two objects. For reconstruction of the objects, the 2D-FRFT is applied to the hologram. The optimal fractional orders for the objects in Figure 4b,c are -0.600 \(\pi/2\) and -0.686 \(\pi/2\), respectively. These two optimal fractional orders of the reconstructed objects give the correct longitudinal positions along z-axis of both objects. These reconstructions show the difference  
Figure 4: **(a)** Simulated hologram considering both phase and opaque objects located in two different planes; **(b)** Reconstructed phase object with optimal order \(\alpha_{x,y}^{opt}=-0.600\)\(\pi/2\) (red arrow); **(c)** Reconstructed opaque object with optimal order \(\alpha_{x,y}^{opt}=-0.686\)\(\pi/2\) (red arrow). between opaque and phase objects: for the pure phase object in Figure 4b, a bright and a dark contour due to the discontinuity are observed at the edge of the object as mentioned in [13]. The intensity inside the reconstructed object can be higher than the continuous background. For the opaque object in Figure 4c, the intensity in the reconstructed object is zero.