Computer Optics and Nanophotonics CALCULATION OF LASER RADIATION DIFFRACTION ON CRYSTAL STRUCTURES BASED ON 3D FOURIER TRANSFORM O.A. Mossoulina1, N.V. Kalinin1, M.S. Kirilenko1,2 1 Samara National Research University, Samara, Russia 2 Image Processing Systems Institute - Branch of the Federal Scientific Research Centre “Crys- tallography and Photonics” of Russian Academy of Sciences, Samara, Russia Abstract. In this paper a calculation of n-dimensional (n = 1,2,3) Fourier trans- form for various structures, including Gauss-Hermite modes and perovskite crystal cells is made. A significant difference in the diffraction pattern on primi- tive crystal cells and on perovskite-type crystals is shown. Keywords: diffraction on crystal structures, n-dimensional Fourier transform, Hermite-Gaussian modes, perovskites. Citation: Mossoulina OA, Kalinin NV, Kirilenko MS. Calculation of laser ra- diation diffraction on crystal structures based on 3D Fourier transform. CEUR Workshop Proceedings, 2016; 1638: 76-82. DOI: 10.18287/1613-0073-2016- 1638-76-82 1 Introduction Crystallography began to be actively developed after the discovery of X-rays by Wil- helm Conrad Roentgen in 1895 [1]. The first X-ray experiment was conducted by Max von Laue in Munich in 1912 [2]. During his work von Laue received experi- mental evidence that X-rays consisted of high-energy particles; the other data showed that X-rays could be waves. Von Laue suggested that if X-rays were waves, they would have rather short wavelengths. In 1913 an important event was the discovery made by William Henry and William Lawrence Bragg that X-rays can be used to study positions of atoms inside a crystal, and most importantly, its three-dimensional structure [3]. The inverse lattice is an important mathematical image, and it finds numerous applications in geometric crys- tallography, in the theory of diffraction and in the structural analysis of crystals, in solid-state physics [4]. At the end of the 20th century a new type of solid substance was discovered, which as well as the crystals has a discrete diffraction pattern, but at the same time has a banned symmetric structure. Such substances are called quasicrystals [5, 6]. Information Technology and Nanotechnology (ITNT-2016) 76 Computer Optics and Nanophotonics Mossoulina OA, Kalinin NV, Kirilenko MS… Additionally, some hybrid organic-inorganic semiconductor materials, for example with perovskite structure [7], recently acquire great importance in such applied fields as photovoltaics [8, 9], where previously crystal silicon was the classical material. The term "perovskite" used for such materials reflects the type of crystal lattice ABX3, corresponding to the structure of the natural mineral perovskite CaTiO3, named after Russian mineralogist L.A. Perovsky [10]. Thus, presently not only the existing (natural) crystals are under investigation, but also there is a rapidly developing area of creating new crystals with unusual proper- ties, and also there is an urgent task of investigating the properties of hybrid struc- tures. The task of calculating the diffraction on the crystal structures can be reduced to per- forming 3D Fourier transformation [11, 12]. The urgency of implementation of the direct method of calculation of the diffraction pattern on crystal structure is associated with emergence of a wide variety of new artificial crystals - quasicrystals, for which not only three-dimensional, but also two- dimensional, and even one-dimensional structures are relevant [13, 14]. In this article the calculation of n-dimensional (n = 1,2,3) Fourier transform for vari- ous structures, including Gauss-Hermite modes and perovskite crystal cell. 2 Theoretical background Let us consider the n-dimensional Fourier transformation: F (u)   f (x) (u)   f (x) exp  2 ixu  d n x , (1) Rn where f ( x ) is the input function from one variable, F (u ) s the resulting output function, [ f (x)](u) is Fourier transform operator. As input functions let us consider the functions of Hermite-Gauss [15], which are a family of own Fourier transform functions [16, 17]:  x2  x f (x)  ψ m ( x, σ)  A exp    Hm   , (2)  2σ  σ where H m  x  - n-dimensional Hermite polynomial of m-order, A is a normalization coefficient. And also let us consider a superposition of n-dimensional Gaussian beams with which different structures can be approximated, including crystal ones: P  (x  x p ) 2  f (x)   c p exp   , (3) p 1   p2  where x p can be regarded as positions of crystal cell nodes, and  p s effective di- mensions of respective atoms. Information Technology and Nanotechnology (ITNT-2016) 77 Computer Optics and Nanophotonics Mossoulina OA, Kalinin NV, Kirilenko MS… Note that the Fourier transform of the function (3) to the infinite limits can also be calculated analytically: F (u)   P  c p p exp   2 p2 u 2  exp  2 ix p u  . P (4) p 1 From the expression (4) it is obvious that an interference pattern in the far zone will depend on the positions of nodes as well on the size of atoms. 3 The results of modeling This section presents the results of calculation of the n-dimensional (n = 1,2,3) Fouri- er transform for the structures kinds (2) and (3). Table 1 shows the results of calculation for one-dimensional distributions, Table 2 - for two-dimensional, and Table 3 - for three-dimensional. We can see the change in the interference pattern in the diffraction far field depending on the distance between the Gaussian beams, their weight ratios and amounts. Table 1. The spatial spectrum of one-dimensional structures Input distribution Absolute value of Fourier transform Information Technology and Nanotechnology (ITNT-2016) 78 Computer Optics and Nanophotonics Mossoulina OA, Kalinin NV, Kirilenko MS… Table 2. The spatial spectrum of two-dimensional structures Input distribution Absolute value of Fourier trans- form For three-dimensional case visualization was carried out using the software package ParaView. Fig. 1 shows the three-dimensional Hermite-Gaussian modes with m  (2,2,2) and m  (1,2,3) indexes. In the area of spatial spectrum these modes are totally reproduced. Also in the three-dimensional case two primitive crystal cells (Table 3) and the struc- ture of the perovskite type were examined. A significant difference should be noted between the spectral picture of primitive cells and the diffraction on perovskite. Information Technology and Nanotechnology (ITNT-2016) 79 Computer Optics and Nanophotonics Mossoulina OA, Kalinin NV, Kirilenko MS… Table 3. The spatial spectrum of three-dimensional structures Input distribution Absolute value of Fourier trans- form Cubic cell Homocentric cell Perovskite cell There are many serious packages for studies in the field of crystallography, for exam- ple, CRYSTAL, VASP. However, they usually require a certain access and powerful clusters [18]. At the same time there are many open-source software products, such as STEM_CELL and XRayView, allowing to build projections of diffraction patterns for certain crystal structures. Information Technology and Nanotechnology (ITNT-2016) 80 Computer Optics and Nanophotonics Mossoulina OA, Kalinin NV, Kirilenko MS… a) b) Fig. 1. Distribution of 3D Hermite-Gaussian modes (2) with indexes a) m  (2,2,2) , b) m  (1,2,3) Figure 2 shows the model of perovskite structure CaTi04, and Fig. 3 – the diffraction patterns in the direction of the axes (0,0,1) and (1,1,1). a) b) Fig. 2. A model of perovskite structure CaTi04: a) view of a single cell, and b) of the structure obtained by copying. a) b) Fig. 3. The diffraction patterns for perovskite CaTi04 in the direction of the axes a) (0,0,1) and b) (1,1,1). Information Technology and Nanotechnology (ITNT-2016) 81 Computer Optics and Nanophotonics Mossoulina OA, Kalinin NV, Kirilenko MS… 4 Conclusion In this article for approximation of various structures, including crystal ones, superpo- sitions of n-dimensional Gaussian beams were examined. 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