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Astigmatic transformation of a fractional-order edge dislocation
V.V. Kotlyar 1,2, E.G. Abramochkin 3, A.A. Kovalev 1,2, A.G. Nalimov 1,2

IPSI RAS – Branch of the FSRC "Crystallography and Photonics" RAS,
443001, Samara, Russia, Molodogvardeyskaya 151,
Samara National Research University, 443086, Samara, Russia, Moskovskoye Shosse 34;
Lebedev Physical Institute, 443034, Samara, Russia, Novo-Sadovaya 221

 PDF, 2339 kB

DOI: 10.18287/2412-6179-CO-1084

Pages: 522-530.

Full text of article: Russian language.

It is shown theoretically that an astigmatic transformation of an edge dislocation (straight line of zero intensity) of the ν-th order (ν=n+α is a real positive number, n is integer, 0<α<1 is the fractional part of the number) forms at twice the focal length from a cylindrical lens n optical elliptical vortices (screw dislocations) with a topological charge of –1, located on a straight line perpendicular to the edge dislocation. Coordinates of these points are zeros of the Tricomi function. At some distance from these vortices and on the same straight line, another additional vortex with a topological charge of –1 is also generated, which moves to the periphery if α decreases to zero, or approaches n vortices if α tends to 1. In addition, at the periphery in the beam cross-section, a countable number of optical vortices (intensity zeros) are formed, all with a topological charge of –1, which are located on diverging curved lines (such as hyperbolas) equidistant from a straight line on which the main n intensity zeros are located. These "accompanying" vortices approach the center of the beam, following the additional "passenger" vortex, if 0<α<0.5, or move to the periphery, leaving the "passenger" next to the main vortices, if 0.5<α<1. At α=0 and α=1, the "accompanying" vortices are situated at infinity. The topological charge of the entire beam at fractional ν is infinite. The numerical simulation confirms theoretical predictions.

astigmatic transformation, fractional order, edge dislocation, screw dislocation, elliptical optical vortex.

Kotlyar VV, Abramochkin EG, Kovalev AA, Nalimov AG. Astigmatic transformation of a fractional-order edge dislocation. Computer Optics 2022; 46(4): 522-530. DOI: 10.18287/2412-6179-CO-1084.

The work was partly funded by the Russian Science Foundation grant 1819-00595 (Section "Orbital angular momentum") and the Ministry of Science and Higher Education of the Russian Federation within the government project of the FSRC “Crystallography and Photonics” RAS (Section "Numerical modeling").


  1. Abramochkin E, Volostnikov V. Beam transformations and nontransformed beams. Opt Commun 1991; 83(1-2): 123-135. DOI: 10.1016/0030-4018(91)90534-K.
  2. Lu B, Wu P. Analytical propagation equation of astigmatic Hermite-Gaussian beams through a 4x4 paraxial optical systems and their symmetrizing transformation. Opt Laser Technol 2003; 35: 497-504.
  3. Chen YF, Chay CC, Lee CY, Tung JC, Liang HC, Huang KT. Characterizing the propagation evolution of wave patterns and vortex structures in astigmatic transformations of Hermite-Gaussian beams. Laser Phys 2017; 28: 015002.
  4. Abramochkin EG, Razueva EV, Volostnikov VG. Hermite-Laguerre-Gaussian beams in astigmatic optical systems. Proc SPIE 2008; 7009: 70090M. DOI: 10.1117/12.793382.
  5. Bekshaev AYa, Soskin MS, Vasnetsov MV. Transformation of higher-order optical vortices upon focusing by an astigmatic lens. Opt Commun 2004; 241: 237-247.
  6. Bekshaev AYa, Karamoch AI. Astigmatic telescopic transformation of a high-order optical vortex. Opt Commun 2008; 281: 5687-5696.
  7. Zhu K, Zhu J, Su Q, Tang H. Propagation properties of an astigmatic sin-Gaussian beam in strongly nonlocal nonlinear media. Appl Sci 2019; 9: 71.
  8. Huang TD, Lu TH. Large astigmatic laser cavity modes and astigmatic compensation. Appl Phys B 2018; 124: 72.
  9. Pan J, Shen Y, Wan Z, Fu X, Zhang H, Liu Q. Index-tunable structured-light beams from a laser with a intracavity astigmatic mode converter. Phys Rev Appl 2020; 14: 044048.
  10. Kotlyar VV, Kovalev AA, Porfirev AP, Kozlova ES. Three different types of astigmatic Hermite-Gaussian beams with orbital angular momentum. J Opt 2019; 21(11): 115601. DOI: 10.1088/2040-8986/ab42b5.
  11. Kotlyar VV, Kovalev AA, Porfirev AP. Vortex astigmatic Fourier-invariant Gaussian beams. Opt Express 2019; 27(2): 657-666. DOI: 10.1364/OE.27.000657.
  12. Kotlyar VV, Kovalev AA, Porfirev AP. Elliptic Gaussian optical vortices. Phys Rev A 2017; 95(5): 053805. DOI: 10.1103/PhysRevA.95.053805.
  13. Kotlyar VV, Kovalev AA, Porfirev AP. Astigmatic transforms of an optical vortex for measurement of its topological charge. Appl Opt 2017; 56(14): 4095-4104. DOI: 10.1364/AO.56.004095.
  14. Kotlyar VV, Kovalev AA, Porfirev AP. Vortex Hermite-Gaussian laser beams. Opt Lett 2015; 40(5): 701-704. DOI: 10.1364/OL.40.000701.
  15. Bazhenov VYu, Soskin MS, Vasnetsov MV. Screw dislocations in light wavefronts. J Mod Opt 1992; 39(5): 985-990.
  16. Basistiy IV, Soskin MS, Vasnetsov MV. Optical wavefront dislocations and their properties. Opt Commun 1995; 119(5-6): 604-612.
  17. Petrov DV. Vortex-edge dislocation interaction in a linear medium. Opt Commun 2001; 188: 307-312.
  18. Petrov DV. Splitting of an edge dislocation by an optical vortex. Opt Quantum Electron 2002; 34: 759-773.
  19. He D, Yan H, Lu B. Interaction of the vortex and edge dislocation embedded in a cosh-Gaussian beam. Opt Commun 2009; 282: 4035-4044.
  20. Chen H, Wang W, Gao Z, Li W. Splitting of an edge dislocation by a vortex emergent from a nonparaxial beam. J Opt Soc Am B 2019; 36: 2804-2809.
  21. Kotlyar VV, Kovalev AA, Nalimov AG. Converting an nth-order edge dislocation to a set of optical vortices. Optik 2021; 243: 167453. DOI: 10.1016/j.ijleo.2021.167453.
  22. Berry MV. Optical vortices evolving from helicoidal integer and fractional phase steps. J Opt A–Pure Appl Opt 2004; 6: 259-268.
  23. Gbur G. Fractional vortex Hilbert's Hotel. Optica 2016; 3: 222-225.
  24. Alexeyev CN, Egorov YA, Volyar AV. Mutual transformations of fractional-order and integer-order optical vortices. Phys Rev A 2017; 96(6): 063807. DOI: 10.1103/PhysRevA.96.063807.
  25. Abramochkin EG, Volostnikov VG. Spiral-type beams: optical and quantum aspects. Opt Commun 1996; 125(4-6): 302-323. DOI: 10.1016/0030-4018(95)00640-0.
  26. Abramowitz M, Stegun IA. Handbook of mathematical functions: With formulas, graphs, and mathematical tables. National Bureau of Standards; 1965.
  27. Sedletskii AM. Asymptotics of the zeros of degenerate hypergeometric functions [In Russian]. Matematicheskie Zametki 2007; 82(2): 262-271.
  28. Kotlyar VV, Kovalev AA, Abramochkin EG. Kummer laser beams with a transverse complex shift. J Opt 2020; 22(1): 015606. DOI: 10.1088/2040-8986/ab5ef1.

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