# Computer-Generated Holograms Application in Security Printing
## 2 CGH Basics
Most of the known holographic carriers have limited modulation characteristics. Purely phase, purely amplitude, mostly amplitude, binary phase, and binary amplitude are among the most popular. Though the values of function (1) are independent complex values, they must be encoded onto the points of carrier modulation characteristics. In the case of phase carriers, only the values of wrapped phase argument can be kept:  
\[\Phi(u_{k},v_{l})=\arg[A_{\text{ob}}(u_{k},v_{l})], \tag{4}\]  
where \(\arg\{\cdot\}\)--argument operator. CGH obtained in such a way are known as simple kino-forms. The drawback of this approach is high spatial noise and distortions in the restored image scene due to the loss of object beam amplitude information. Iterative methods based on Gerchberg-Saxton algorithms [9, 10, 11], direct search with random trajectory method [12], and optimal projection algorithms [13] provide the decisions of kinoform fringe pattern function optimization problems (4) that yield a rise of restored scene quality:  
\[\Phi_{\text{opt}}(u_{k},v_{l})=\underset{\Phi_{j}(u_{k},v_{l})\in\Omega}{ \operatorname{argmin}}\big{[}\sigma\big{\{}a_{j},\ a_{0}\big{\}}\big{]}, \tag{5}\]  
where \(\underset{x\in B}{\operatorname{argmin}}[A(x)]\) is the minimization operator of value \(A\) of argument \(x\) within a bounded set B; \(\sigma\big{\{}a_{j},\ a_{0}\big{\}}\) is a similarity measure between spatial amplitudes \(a_{0}\) that is the object field point cloud amplitude function, and \(a_{j}\) is the object amplitude function restored by kinoform \(\Phi_{j}(u_{k},v_{l})\) obtained on \(j\)-th search iteration, and \(\Omega\) is the search rule specific for each of the mentioned iterative methods. The basic limitation of kinoforms is the loss and distortion of the object wave complex amplitude phase argument. This causes significant problems when the formation of 3D scenes is required. Additionally, the implementation of this method requires significant computational burden as soon as iterative forward and backward transformations of the object field within object space and the CGH plane are needed. One of the most promising approaches to operatively realize complex beam functions using a carrier with modulation characteristics limited to amplitude-only or phase-only types is the implementation of the bipolar intensity method, which simulates the classical off-axis Leith-Upatnieks scheme [14]. In that case, a plane wave unit amplitude reference beam model is included into record-simulating algorithms and the CGH fringe pattern function can be found as:  
\[\Gamma(u_{k},v_{l})=\text{Real}\Big{[}A_{\text{ob}}(u_{k},v_{l})\cdot e^{-j \frac{\text{i}\pi}{\lambda}[u_{k}\sin\alpha+v_{l}\sin\beta]}\Big{]}+B \tag{6}\]  
where \(\alpha\) and \(\beta\) are reference beam directional angles in the record scheme model, and \(B\) is constant bias term, which in the case of amplitude carrier must be selected to make \(\Gamma(u_{k},v_{l})\) non-negative. Bipolar intensity method implementation provides a high diffraction efficiency CGHs capable of restoring amplitude and complex-valued data pages in archival holographic data storage systems [2; 6], and 3D scenes [1]. In the case of plane images, application of random and pseudorandom phase masks allow obtaining high diffraction efficiency of holograms. However, the addition of an off-axis reference beam widens the spatial frequency spectrum of fringe pattern, limits the space bandwidth product of discrete carriers, and produces conjugate images in the observation plane. Additionally, in the case when a phase-only carrier is used the presence of high diffraction orders, it causes distortions in the restored image. High quality restoration of an object image without an appearance of a conjugate image can be performed with phase-only carriers and application of the double-phase coding technique [15; 16]. In this case, discrete complex values of \(A_{\text{ob}}(u_{k},v_{l})\) are encoded by two phase-only discrete values of the carrier. \[\Gamma\big{(}u_{k},v_{\ \hat{l}}\big{)}=\left\{