=Paper=
{{Paper
|id=Vol-1638/Paper10
|storemode=property
|title=Calculation of laser radiation diffraction on crystal structures based on 3d Fourier transform
|pdfUrl=https://ceur-ws.org/Vol-1638/Paper10.pdf
|volume=Vol-1638
|authors=Olga A. Mossoulina,Nikita V. Kalinin,Mikhail S. Kirilenko
}}
==Calculation of laser radiation diffraction on crystal structures based on 3d Fourier transform==
https://ceur-ws.org/Vol-1638/Paper10.pdf
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].
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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.
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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
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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.
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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.
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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).
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4 Conclusion
In this article for approximation of various structures, including crystal ones, superpo-
sitions of n-dimensional Gaussian beams were examined. Using n-dimensional (n =
1,2,3) Fourier transform a calculation of the diffraction pattern for different structures
was performed, including Gauss-Hermite modes and perovskite crystal cells.
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