Amplitude and polarization transformations of the Bessel beam as it passes through an anisotropic crystal perpendicular to the axis of the crystal

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

2. Theoretical analysis

d j s jt j R2j   exp ik   d 

j Rj  dx dy , s jt j  where the indices correspond to the ordinary (j = 1) and extraordinary (j = 2) waves, d1  d2  o , s1  t1  1 , s2  t2  o / e .

For transverse (x- and y- components): ( 1 ) e1x ,   1, e1y ,   0, e2x ,   , e2 y ,   2  o . w1 , T  1,   ,  o  2      w2 , T   0,   

1 . o  2  

  1c     1c   where

R1  R2  oo ((vuRR11yx)) ,, 11cc  o (uR1 x) , o (v R1 y) , (u  x)2  (v  y)2  z2 , e o (u  x)2  o (v  y)2  z2 .

e

3. Results of numerical simulation

Similar results can be obtained if the crystal axis is directed along the Ox axis.

During the experiment, the axicon was used. The scheme of the axicon’s work is shown in Fig. 1 ( 2 ) ( 3 ) ( 4 ) ( 5 )

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: n0 = 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 h1 = 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. Component General х у

4. Conclusion

Acknowledgments References

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.

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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