## 3 Methods for Suppressing the Twin-Image Effect
### Complex Field Amplitude Reconstruction Method Based on the Gerchberg-Saxton Algorithm  
The complex field amplitude reconstruction method based on the Gerchberg-Saxton algorithm, which is the iterative calculation of the complex field amplitude (instead of the registered intensity distribution) in the recording plane of the digital hologram, has various implementations and is described in many studies, for example, [14; 15; 16; 17]. This study considers the case when the intensity distributions \(I_{1}(x,y)\) and \(I_{2}(x,y)\) of two digital holograms are recorded at distances z\({}_{1}\) and z\({}_{2}\) from the particle (Figure 5) [15]. Figure 4: Holographic images of particles illustrating the twin-image effect: (**a**) round model particle, (**b**) Daphnia individual, (**c**) three square model particles. Figure 5: Recording of two digital holograms of one particle at different distances from it to implement the Gerchberg–Saxton algorithm. Red arrows point the direction of laser light propagation. Prior to the iteration, it is assumed that the complex field amplitude in the plane of the first hologram \(\mathrm{U_{1}(x,y,z_{1})}=\sqrt{\mathrm{I_{1}(x,y)}}\), i.e., the phase of the complex field amplitude, equals 0. At the first stage of the iteration algorithm, knowing the distribution \(\mathrm{U_{1}(x,y,z_{1})}\) in the plane \(z_{1}\), the complex amplitude of the field \(\mathrm{U_{z_{2}}(x,y,d)}\) is calculated using a diffraction integral after its propagation from the plane at a distance \(z_{1}\) into the plane at a distance \(z_{2}\):  
\[\mathrm{U_{z_{1}\to z_{2}}(x,y,d)}=\exp(\mathrm{ikd})\mathrm{U_{z_{1}}(x,y,z_{ 1})}\otimes\mathrm{h_{d}(x,y)}=|\mathrm{U_{z_{2}}(x,y,d)}|\exp(\mathrm{i} \varphi_{2}(x,y)) \tag{4}\]  
where \(\varphi_{2}(x,y)\)--phase of the complex field amplitude in the plane \(z_{2}\) and \(d=z_{2}-z_{1}\). At the second stage, we calculate the module of the complex field amplitude in the plane at a distance \(z_{2}\) as the square root of the intensity recorded in this plane \(|\mathrm{U_{z_{2}}(x,y,d)}|=\sqrt{\mathrm{I_{2}(x,y)}}\), and use the phase from the result of the previous calculation (Formula (4)):  
\[\mathrm{U_{z_{2}}(x,y,z_{2})}=\sqrt{\mathrm{I_{2}(x,y)}}\exp(\mathrm{i}\varphi _{2}(x,y)). \tag{5}\]  
At the third stage, knowing the complex amplitude of the field \(\mathrm{U_{z_{2}}(x,y,z_{2})}\) (Formula (5)), similar to the first stage, we calculate the complex amplitude of the field \(\mathrm{U_{z_{1}}(x,y,-d)}\) after its propagation from the plane at a distance \(z_{2}\) into the plane at a distance \(z_{1}\):  
\[\mathrm{U_{z_{2}\to z_{1}}(x,y,-d)}=\exp(-\mathrm{ikd})\mathrm{U_{z_{2}}(x,y, z_{1})}\otimes\mathrm{h_{-d}(x,y)}=|\mathrm{U_{z_{1}}(x,y,-d)}|\exp(\mathrm{i} \varphi_{1}(x,y)). \tag{6}\]  
At the fourth stage, using the module of the complex field amplitude in the plane at a distance \(z_{1}\) calculated as \(|\mathrm{U_{z_{1}}(x,y,d)}|=\sqrt{\mathrm{I_{1}(x,y)}}\), and the phase \(\varphi_{1}(x,y)\) from the Formula (6), we obtain  
\[\mathrm{U_{z_{1}}(x,y,z_{1})}=\sqrt{\mathrm{I_{1}(x,y)}}\exp(\mathrm{i} \varphi_{1}(x,y)). \tag{7}\]  
As a result of one iteration consisting of four stages, we obtain complex amplitudes of the fields \(\mathrm{U_{z_{1}}(x,y,z_{1})}\) and \(\mathrm{U_{z_{2}}(x,y,z_{2})}\) instead of the intensity distributions \(\mathrm{I_{1}(x,y)}\) and \(\mathrm{I_{2}(x,y)}\) of recorded digital holograms. Knowing the complex field amplitude in the recording plane, we can calculate the particle transmittance \((1-\mathrm{a}(x,y))\):  
\[ \begin{split}\exp(-\mathrm{ikz_{2}})\mathrm{U_{z_{2}}(x,y,z_{2}) }\otimes\mathrm{h_{-z_{2}}(x,y)}=\exp(-\mathrm{ikz_{2}})\exp(\mathrm{ikz_{2}} )\big{(}\mathrm{T^{\prime}(x,y)}\otimes\mathrm{h_{z_{2}}(x,y)}\big{)}\otimes \mathrm{h_{-z_{2}}(x,y)}\\ =\mathrm{T^{\prime}(x,y)}\otimes(\mathrm{h_{z_{2}}(x,y)}\otimes \mathrm{h_{-z_{2}}(x,y)})=\mathrm{T^{\prime}(x,y)}\otimes\delta(x,y)=\mathrm{ T^{\prime}(x,y)}=1-\mathrm{a^{\prime}(x,y)}.\end{split}