An analysis of the spatial spectrum of Hermite-Gauss beams with embedded optical vortex
Ustinov A.V.


Image Processing Systems Institute оf RAS – Branch of the FSRC “Crystallography and Photonics” RAS, Samara, Russia

Full text of article: Russian language.


This work deals with a detailed theoretical study of Hermite-Gauss beams with embedded vortical phase passing through a lens. Special attention is given to the location of vortical phase singularities in the focal plane. Sufficient conditions for the intensity null in the focal plane centre are found. The theoretical analysis is confirmed by numerical calculations.

Hermite-Gauss beam; phase singularity; trigonometric transforms.

Ustinov AV. An analysis of the spatial spectrum of Hermite-Gauss beams with embedded optical vortex. Computer Optics 2017; 41(5): 617-625. DOI: 10.18287/2412-6179-2017-41-5-617-625.


  1. Nye JF, Berry MV. Dislocations in wave trains. Proc R Soc Lond A 1974; 336: 165-190. DOI: 10.1098/rspa.1974.0012.
  2. Coullet P, Gil G, Rocca F. Optical vortices. Opt Commun 1989; 73: 403-408. DOI: 10.1016/0030-4018(89)90180-6.
  3. Bazhenov VY, Vasnetsov MV, Soskin MS. Laser-beam with screw dislocations in the wavefront. JETP Lett 1990; 52(8): 429-431.
  4. Khonina SN, Kotlyar VV, Shinkarev MV, Soifer VA, Uspleniev GV. The rotor phase filter. J Mod Opt 1992; 39(5): 1147-1154. DOI: 10.1080/09500349214551151.
  5. Soskin MS, Vasnetsov MV. Singular optics. In book: Wolf E, ed. Progress in Optics. Chapter 4. Amsterdam, North Holland: Elsevier Science; 2001: 219-276. ISBN: 978-0-444509086.
  6. Andrews DL, ed. Structured light and its applications: An introduction to phase-structured beams and nanoscale optical forces.  Burlington, San Diego, London: Elsevier Inc.; 2008. ISBN: 978-0-12-374027-4.
  7. Kotlyar VV, Soifer VA, Khonina SN. Rotation of Gauss-Laguerre multimodal light beams in free space. Tech Phys Lett 1997; 23(9): 657-658.
  8. Khonina SN, Kotlyar VV. Bessel-mode formers. Proc SPIE 1994; 2363: 184-190. DOI: 10.1117/12.199633.
  9. Arlt J, Dholakia K. Generation of high-order Bessel beams by use of an axicon. Opt Commun 2000; 177(1-6): 297-301. DOI: 10.1016/S0030-4018(00)00572-1.
  10. Kotlyar VV, Skidanov RV, Khonina SN, Soifer VA. Hypergeometric modes. Opt Lett 2007; 32(7): 742-744. DOI: 10.1364/OL.32.000742.
  11. Khonina SN, Kotlyar VV, Wang Ya. Diffractive optical element matched with Zernike basis. Pattern Recognition and Image Analysis 2001; 11(2): 442-445.
  12. Abramochkin EG, Volostnikov VG. Beam transformation and nontransformed beams. Opt Commun 1991; 83(1-2): 123-135. DOI: 10.1016/0030-4018(91)90534-K.
  13. Beijersbergen MW, Allen L, Van der Veen HE, Woerdman JP. Astigmatic laser mode converters and transfer of orbital angular momentum. Opt Commun 1993; 96: 123-132. DOI: 10.1016/0030-4018(93)90535-D.
  14. Mazilu M, Baumgartl J, Cizmar T, Dholakia K. Accelerating vortices in Airy beams. Proceedings of SPIE 2009; 7430: 74300C. DOI: 10.1117/12.826372.
  15. Dai HT, Liu YJ, Luo D, Sun XW. Propagation dynamics of an optical vortex imposed on an Airy beam. Opt Lett 2010; 35(23): 4075-4077. DOI: 10.1364/OL.35.004075.
  16. Chen R-P, Chew K-H, He S. Dynamic control of collapse in a vortex Airy beam. Scientific Reports 2013; 3: 1406. DOI: 10.1038/srep01406.
  17. Kotlyar VV, Kovalev AA, Porfirev AP. Vortex Hermite–Gaussian laser beams. Opt Lett 2015; 40(5): 701-704. DOI: 10.1364/OL.40.000701.
  18. Porfirev AP, Khonina SN. Simple method for efficient reconfigurable optical vortex beam splitting. Opt Express 2017; 25(16): 18722-18735. DOI: 10.1364/OE.25.018722.
  19. Hacyan S, Jáuregui R. Evolution of optical phase and polarization vortices in birefringent media. J Opt A: Pure Appl Opt 2009; 11(8): 085204. DOI: 10.1088/1464-4258/11/8/085204.
  20. Khonina SN, Paranin VD, Ustinov AV, Krasnov AP. Astigmatic transformation of Bessel beams in a uniaxial crystal. Optica Applicata 2016; XLVI(1): 5-18. DOI: 10.5277/oa160101.
  21. Kotlyar VV, Khonina SN, Almazov AA, Soifer VA, Jefimovs K, Turunen J. Elliptic Laguerre-Gaussian beams, J Opt Soc Am A 2006; 23: 43-56. DOI: 10.1364/JOSAA.23.000043.
  22. Prudnikov AP, Brychkov YuA, Marichev OI. Integrals and Series: Vol. 1: Elementary Functions; Vol. 2: Special Functions. New York: CRC Press; 1986.

© 2009, IPSI RAS
Institution of Russian Academy of Sciences, Image Processing Systems Institute of RAS, Russia, 443001, Samara, Molodogvardeyskaya Street 151; E-mail: ; Phones: +7 (846 2) 332-56-22, Fax: +7 (846 2) 332-56-20