## 3 Methods for Suppressing the Twin-Image Effect  
Let us consider the in-line scheme (Figure 3) to record a hologram of a certain particle that is illuminated by a flat wave with unit amplitude for simplicity. The amplitude transmission in a particle plane \(\left(x_{0},y_{0}\right)\):  
\[\mathrm{T}\left(x_{0},y_{0}\right)=1-\mathrm{a}\left(x_{0},y_{0}\right), \tag{1}\]  
where \(\mathrm{a}\left(x_{0},y_{0}\right)\)--function describing a particle (amplitude absorption of a particle) [2]. Note that the amplitude transmission \(1-\mathrm{a}\left(x_{0},y_{0}\right)\) under illumination with a flat wave with unit amplitude up to a constant is equal to the complex amplitude of the field in the plane \(\left(x_{0},y_{0}\right)\) directly behind the particle. A CCD or CMOS camera is placed in the interference region of the reference and object waves, in the recording plane of the digital hologram \(\left(x_{1},y_{1}\right)\). The complex amplitude of the field in the recording plane \(\left(x_{1},y_{1}\right)\) can be represented by a diffraction integral:  
\[ \begin{array}{l}\mathrm{U}(x_{1},y_{1},z)=\iint\limits_{\infty}^{\infty} \mathrm{T}\left(x_{0},y_{0}\right)\frac{\exp(\mathrm{i}kx)}{1\mathrm{x}z}\cdot \exp\left(\frac{\mathrm{i}k}{2\mathcal{L}}\left[\left(x_{1}-x_{0}\right)^{2}+ \left(y_{1}-y_{0}\right)^{2}\right]\right)\mathrm{d}x_{0}\mathrm{d}y_{0}\\ =\exp(\mathrm{i}kx)\iint\limits_{\infty}^{\infty}\mathrm{T}\left(x_{0},y_{0} \right)\mathrm{h}_{\mathrm{z}}\left(x_{1}-x_{0},y_{1}-y_{0}\right)\mathrm{d}x _{0}\mathrm{d}y_{0}=\exp(\mathrm{i}kx)(\mathrm{T}\otimes\mathrm{h}_{\mathrm{ z}}(x,y)),\end{array}  \tag{2}\]  
where z--distance from the particle plane to the recording plane, \(\mathrm{k}=\frac{2\pi}{\lambda}\)--wave number, \(\lambda\)--wavelength of the illuminating radiation, \(\mathrm{h}_{\mathrm{z}}(x,y)=\frac{1}{\mathrm{i}\lambda z}\cdot\exp\left(\frac {\mathrm{i}k}{2\mathcal{L}}\left(x^{2}+y^{2}\right)\right)\)--convolution kernel (Fresnel function), and \(\otimes\) denotes the convolution itself. The intensity distribution recorded by the CCD camera is \(\mathrm{I}_{\mathrm{H}}(x_{1},y_{1})=\mathrm{U}(x_{1},y_{1},z)\cdot\mathrm{U}^ {*}(x_{1},y_{1},z)\). A particle image is reconstructed numerically from a digital hologram using the diffraction integral:  
\[\mathrm{U}(x_{2},y_{2},z)=\iint_{\infty}^{\infty}\mathrm{t}(x_{1},y_{1})\frac {\exp(\mathrm{i}kx)}{\mathrm{i}\mathrm{z}z}\cdot\exp\left(\frac{\mathrm{i}k}{ 2\mathcal{L}}\left[\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2} \right]\right)\mathrm{d}x_{1}\mathrm{d}y_{1}. \tag{3}\]  
Here, \(\mathrm{t}(x_{1},y_{1})\) is the amplitude transmission of a hologram, which, in the case of ideal hologram recording, is equal, up to a constant, to the recorded intensity \(\mathrm{I}_{\mathrm{H}}(x_{1},y_{1})\), and when the hologram is illuminated with a plane wave with unit amplitude, it equals, up to a constant, to the complex amplitude of the reconstructed wave immediately behind the  
Figure 3: Formation of a twin image. The green circle marks the boundary area of the real image. Red arrows point the direction of laser light propagation. hologram in the plane \((x_{1},y_{1})\). In digital holography, the hologram illumination and image reconstruction are modeled numerically. In both optical and digital holography, a reconstructed wave in a plane located at a distance z from the hologram represents an overlapping of two coherent waves (Figure 3): a wave that is a complex conjugate object wave and forms a real image of a particle, and a wave that propagates from a virtual image located at a distance 2z from the real image. The result of this interaction is a system of fringes overlapping a sharp image of a particle (Figure 4)--the so-called twin-image effect.