=Paper=
{{Paper
|id=Vol-1900/paper13
|storemode=property
|title=Amplitude and polarization transformations of the Bessel beam as it passes through an anisotropic crystal perpendicular to the axis of the crystal
|pdfUrl=https://ceur-ws.org/Vol-1900/paper13.pdf
|volume=Vol-1900
|authors=Anna V. Glazkova,Mariya V. Zablovskaya,Vladimir V. Podlipnov
}}
==Amplitude and polarization transformations of the Bessel beam as it passes through an anisotropic crystal perpendicular to the axis of the crystal ==
https://ceur-ws.org/Vol-1900/paper13.pdf
Amplitude and polarization transformations of the Bessel beam as it passes
through an anisotropic crystal perpendicular to the axis of the crystal
A.V. Glazkova1, M.V. Zablovskaya1, V.V. Podlipnov1,2
1
Samara National Research University, 34 Moskovskoe Shosse, 443086, Samara, Russia
2
Image Processing Systems Institute – Branch of the Federal Scientific Research Centre “Crystallography and Photonics” of Russian Academy of Sciences, 151
Molodogvardeyskaya st., 443001, Samara, Russia
Abstract
A comparative numerical calculation of the propagation of a zero-order Bessel laser beam in a uniaxial crystal perpendicular to its axis is
performed using the Rayleigh-Sommerfeld integral operator, generalized for an anisotropic environment. Numerical simulation is performed
with a different type of beam polarization and different characteristics of the Bessel beam. Patterns of the beam intensity during the passing of
different distances in the crystal are obtained, showing the degree of astigmatic transformation, which makes it possible to determine the
conditions under which the greatest astigmatic distortion of the beams occurs. The above analysis can be useful in practice for determining the
anisotropy characteristics of a crystal.
Keywords: diffraction axicon; birefringent crystal; polarization transformations; amplitude transformations, Bessel beams; astigmatism
1. Introduction
Optical devices are becoming more and more interesting and practical. They allow to transform certain properties of
electromagnetic radiation into others. Most often, modal transformations (from the fundamental mode to higher order
distributions) and polarization (from homogeneous linear polarization to more complex ones) are required. One of the tools of
such transformations are anisotropic crystals. The propagation of laser modes with a high numerical aperture in an environment
with strong anisotropy leads to complex polarization-mode transformations [1-6].
In particular, when propagating along the crystal axis, the spin angular momentum is transformed, which has a circularly
polarized beam at the orbital angular momentum [7-13]. It was shown in [6, 7, 11-13] that when propagating along the crystal
axis, nonparaxial Bessel beams undergo a periodic change in intensity, corresponding to a transformation into a higher-order
beam. In publications [14-20], polarization transformations of beams focused along the crystal axis were considered.
The propagation of various types of laser beams perpendicular to the axis of the crystal was investigated in [21-26]. The most
interesting transformations were observed for Bessel beams [16, 21, 24, 27], since in this case there is a visually pronounced
astigmatic distortion of the ring structure of the beam. A similar distortion can be observed with oblique incidence of a plane
wave on a diffraction axicon [28-30], and also with a cylindrical lens [31]. This analogy was noted in [24], and the analytical
basis for such an effect was given in [27].
In this paper, the effect of the astigmatic transformation of Bessel beams propagating perpendicular to the crystal axis is
studied in detail on the basis of numerical simulation. The calculation was carried out using the Rayleigh-Sommerfeld integral
operator, generalized for an anisotropic environment [32, 33]. Numerical simulation is performed for different types of beam
polarization and different characteristics of the Bessel beam. The formation of Bessel beams [34-37] was carried out with the
diffraction axicon with different period of the radial lattice. The effect of the relative position of the polarization plane of the
radiation and the c-axis of the crystal on the intensity distributions formed in different vector components of ordinary and
extraordinary beams is investigated. Patterns of the beam intensity are obtained during the passing of different distances in the
crystal, showing the degree of astigmatic transformation, which makes it possible to determine the conditions under which the
greatest astigmatic distortion of the beams occurs. The above analysis can be useful in practice for determining the anisotropy
characteristics of a crystal.
2. Theoretical analysis
Consider an anisotropic crystal whose axis is oriented perpendicular to the propagation axis and coincides with the Oy axis. In
this case, the field propagation in a crystal with dielectric permittivities, (ordinary and extraordinary) can be described by an
expression similar to the Rayleigh-Sommerfeld integral [32, 33]:
2z 2 d j s jt j
dj
e jc , jc w j jc , jc E ( x, y, 0)
2 j
T
E(u, v, z ) exp ik R j dx dy , (1)
j 1
Rj 2
s jt j
d d 2 o s1 t1 1 s2 t2 o / e
where the indices correspond to the ordinary (j = 1) and extraordinary (j = 2) waves, 1 , , .
For transverse (x- and y- components):
3rd International conference “Information Technology and Nanotechnology 2017” 60
Computer Optics and Nanophotonics / A.V. Glazkova, M.V. Zablovskaya, V.V. Podlipnov
e1x , 1,
e1 y , 0,
e2 x , ,
e2 y , 2 o .
(2)
w1 , 1,
T
,
o 2
1
w 2 , 0, .
T
o 2
(3)
(u x) (u x)
1c o R , 1c o R ,
1 1
( v y) ( v y)
, ,
1c o
R1
1c o
R1
(4)
where
R1 (u x) 2 (v y ) 2 z 2 ,
e
R2 (u x) 2 o (v y ) 2 z 2 .
o e
(5)
Similar results can be obtained if the crystal axis is directed along the Ox axis.
3. Results of numerical simulation
During the experiment, the axicon was used. The scheme of the axicon’s work is shown in Fig. 1
Fig. 1. The scheme of the axicon’s work.
In order to carry out the simulation as an anisotropic medium, a lithium niobate crystal of the X-cut was chosen in this study,
the dielectric constant of which is ε0 = 5.2273505956, εe = 4.8517551289. The refractive indices of this crystal are: n 0 =
2.28634, ne = 2.20267. For the formation of zero-order Bessel beams, diffraction axicons with periods d1 = 1.2 μm, d2 = 2 μm, d3
= 4 μm were used and illuminated with light polarized linearly along the OY axis with a wavelength of = 632.8 nm. We also
compared the results of the transformation for different crystal thicknesses, which were chosen h 1 = 1047 μm and h2 = 843 μm.
To analyze the transformation of Bessel beams with axicons, the results of the simulation were presented in the form of patterns
of light distribution of propagating beams separately for polarized light along OX, separately for OY and their superposition.
The results of the simulation are presented in Table 1.
It can be noted that the picture of the Y component almost does not differ from the superposition picture of the X and Y
components, which means that the X component has a negligible intensity, and the linearly polarized light at the exit from the
lithium niobate crystal has not changed its polarization.
As can be seen from the modeled intensity distribution maps of lithium niobate transformed into an anisotropic lithium
crystal by Bessel beams, the beams formed by axicons with the minimal period are subjected to the strongest astigmatic
distortions. With an increase in crystal thickness, the degree of astigmatism increases in proportion to the propagation length.
When analyzing patterns of light intensity distribution at the output of an anisotropic crystal for linearly polarized light along
the Y axis, with circular polarization, polarization rotated through an angle of 45 ° about the X axis, the above-described
character of the Bessel beam transformation is preserved.
3rd International conference “Information Technology and Nanotechnology 2017” 61
Computer Optics and Nanophotonics / A.V. Glazkova, M.V. Zablovskaya, V.V. Podlipnov
Table 1. Patterns of propagation of Bessel beams formed with axicons under illumination by light polarized along the OY axis through an anisotropic X-cut
crystal.
d1=1,2 μm d2=2 μm d3=4 μm
Compo- h1=1,047 mm h2=0,843 mm h1=1,047 mm h2=0,843 mm h1=1,047 mm h2=0,843 mm
nent
General
х
у
4. Conclusion
In the work, to analyze the dependence of the propagation of the zero-order Bessel beam on the polarization angle, on the
period and the radius of the axicon, we used the calculation with the Rayleigh-Sommerfeld integral operator generalized for an
anisotropic medium. The Bessel beams formed by an axicon with the smallest period and passing through an anisotropic Crystal
at the greatest distance. The described regularities can be used in practice to determine the degree of anisotropy or the exact
thickness of the crystal cuts.
Acknowledgments
This work was supported by the Ministry of Education of the Russian Federation., by the Russian Foundation for Basic
Research (RFBR grants 16-07-00825, 16-29-11698 ofi_m and 16-07-00494 a) and by the grant from the President of the
Russian Federation (project no. MD- 5205.2016.9).
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