# Computer-Generated Holograms Application in Security Printing
## 2 CGH Basics  
The main subject of computer holography is CGH, a real-world diffractive element with the fringe pattern model being numerically calculated and realized by displaying with a spatial light modulator (SLM) or printed on any type of surface or volumetric media. In order to restore the object imbedded onto CGH, a coherent or partially coherent optical setup is required. The advantages of computer holography are the absence of requirement of complex interferometric setup application for obtaining the real-world hologram and the possibility to record physically inexistent objects that are only presented as digital models. Depending on the optical setup that is being used to read out the encoded object image, an appropriate model of hologram record setup must be simulated. The main problem in this process is representation (rendering) of the object wave complex amplitude function in the plane of the holographic carrier [1]. Using a model of the object, which can be represented as set of illuminating sources with amplitudes \(\{a_{n}\}\), the object beam function can be found as the sum of elementary waves from each element of object model (Figure 1). In the case of a point cloud model, each element illuminates spherical wave, and the object beam function can be found as (see also Figure 1):  
\[A_{\text{ob}}(u_{k},v_{l})=\sum_{n=0}^{N-1}\frac{a_{n}}{r_{kln}}e^{j(\frac{2\pi }{\Lambda}r_{kln}+\phi_{n})}, \tag{1}\]  
where \(r_{kln}\) is the distance between illuminating point with coordinates \((x_{n},y_{n};z_{n})\) to CGH point \((u_{k},v_{l};0)\); \(k=0,\ldots K\) and \(l=0,\ldots L\) are integer numerators of object wave discrete model elements along \(U\) and \(V\) axes correspondingly, e.g.,  
\[r_{kln}=\sqrt{\left(u_{k}-x_{n}\right)^{2}+\left(v_{l}-y_{n}\right)^{2}+z_{n} ^{2}} \tag{2}\]  
Figure 1: Schematic illustration of reference beam rendering process. In the case of large values of \(N\), \(K\), and \(L\), a direct calculation of Equation (1) requires significant computational burden. Different approaches are known to speed up this process. Look-up table methods use precalculation of spherical waves basic set and storing it in memory [7,8]. Those methods require a significant amount of memory usage and a lot of calculations must be performed. In the case of plane 2D objects, such as holographic memory data pages or with the use of planar slicing of 3D objects, model algorithms based on discretized forms of scalar diffraction theory integral transforms can be implemented in order to speed-up calculations. Depending on the propagation distance between CGH and image planes, discrete Fourier transformation for Fraunhofer diffraction zones (3a), discrete direct Fresnel transformation for far Fresnel zones (3b), or discrete angular spectrum propagation for short distances (3c) can be used in object beam rendering algorithms. These algorithms can be implemented in matrix form using the discrete fast Fourier transform (DFFT) operation as follows:  
\[\mathbf{A}_{\text{ob}}=\mathcal{F}[\mathbf{a}_{\text{ob}}], \tag{3a}\] \[\mathbf{A}_{\text{ob}}=\mathcal{F}[\mathbf{a}_{\text{ob}}\circ\mathbf{C}_{ \text{ob}}]\circ\mathbf{C}_{\text{h}},\] (3b) \[\mathbf{A}_{\text{ob}}=\mathcal{F}\Big{[}\mathcal{F}^{-1}[\mathbf{a}_{\text{ob }}]\circ\mathbf{C}_{\text{ss}}\Big{]}, \tag{3c}\]  
where \(\mathcal{F}\)--DFFT operation, \(\mathcal{F}^{-1}\)--inversed DFFT operation, and \(\mathbf{C}_{\text{ob}}\), \(\mathbf{C}_{\text{h}}\), and \(\mathbf{C}_{\text{ss}}\) are spatial chirp matrices:  
\[\mathbf{C}_{\text{ob}}=\mathbf{C}_{\text{as}}=\exp\left(j\pi\frac{\mu\big{(}m ^{2}+n^{2}\big{)}}{N}\right);\mathbf{C}_{\text{h}}=\exp\left(j\pi\frac{\big{(} m^{2}+n^{2}\big{)}}{\mu N}\right);\mu=\frac{\lambda z_{0}}{N\Delta_{h}}\]  
where \(\lambda\) is wavelength; \(z_{0}\) is propagation distance; \(N\) is the total size of the calculation field in pixels; and \(\Delta_{h}\) is the discretization step in the CGH plane. The next step of the CGH application process is the synthesis of the CGH fringe pattern discrete model.