Orbital angular momentum of an arbitrary axisymmetric light field after passing through an off-axis spiral phase plate
Kotlyar V.V., Kovalev A.A.

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


We obtain a simple formula for the relative total orbital angular momentum (OAM) of a paraxial light beam with arbitrary rotationally symmetric complex amplitude passed through a spiral phase plate (SPP) whose center is shifted from the optical axis. The formula shows that the OAM equals zero if the incident beam is bounded by an aperture and the SPP center is outside this aperture. For the incident beam bounded by an annular aperture, there is another interesting consequence of the obtained expression. The total OAM of such a beam is the same regardless of the position of the SPP center within the shaded circle of the aperture. Thus, it would be appropriate to illuminate the SPP by beams with an annular intensity distribution, since in this case an inaccurate alignment of the SPP center and the center of the annular intensity distribution does not affect the total OAM of the beam. We also obtain an expression for the OAM density of such a beam in the initial plane.

orbital angular momentum, shifted spiral phase plate, rotationally-symmetric light beam, circular aperture, annular aperture.

Kotlyar VV, Kovalev AA. Orbital angular momentum of an arbitrary axisymmetric light field after passing through an off-axis spiral phase plate. Computer Optics 2018; 42(2): 212-218. DOI: 10.18287/2412-6179-2018-42-2-212-218.


  1. Mair A, Vaziri A, Weihs G, Zeilinger A. Entanglement of the orbital angular momentum states of photons. Nature 2001; 412: 313-316. DOI: 10.1038/35085529.
  2. Vaziri A, Weihs G, Zeilinger A. Superpositions of the orbital angular momentum for applications in quantum experiments. J Opt B: Quant Semicl Opt 2002; 4(2): S47-S51. DOI: 10.1088/1464-4266/4/2/367.
  3. Chen Q-F, Shi B-S, Zhang Y-S, Guo G-C. Entanglement of the orbital angular momentum states of the photon pairs generated in a hot atomic ensemble. Phys Rev A 2008; 78(5): 053810. DOI: 10.1103/PhysRevA.78.053810.
  4. Janicijevic L, Topuzoski S. Gaussian laser beam transformation into an optical vortex beam by helical lens. J Mod Opt 2016; 63(2): 164-176. DOI: 10.1080/09500340.2015.1085106.
  5. Ricci F, Löffler W, van Exter MP. Instability of higher-order optical vortices analyzed with a multi-pinhole interferometer. Opt Express 2012; 20(20): 22961-22975. DOI: 10.1364/OE.20.022961.
  6. Kumar A, Vaity P, Singh RP. Crafting the core asymmetry to lift the degeneracy of optical vortices. Opt Express 2011; 19(7): 6182-6190. DOI: 10.1364/OE.19.006182.
  7. Dennis MR. Rows of optical vortices from elliptically perturbing a high-order beam. Opt Lett 2006; 31(9): 1325-1327. DOI: 10.1364/OL.31.001325.
  8. Kotlyar VV, Khonina SN, Almazov AA, Soifer VA, Jefimovs K, Turunen J. Elliptic Laguerre–Gaussian beams. J Opt Soc Am A 2006; 23(1): 43-56. DOI: 10.1364/JOSAA.23.000043.
  9. Li Z, Zhang M, Liang G, Li X, Chen X, Cheng C. Generation of high-order optical vortices with asymmetrical pinhole plates under plane wave illumination. Opt Express 2013; 21(13): 15755-15764. DOI: 10.1364/OE.21.015755.
  10. Kotlyar VV, Kovalev AA, Soifer VA. Asymmetric Bessel modes. Opt Lett 2014; 39(8): 2395-2398. DOI: 10.1364/OL.39.002395.
  11. Kovalev AA, Kotlyar VV, Porirev AP. Asymmetric Laguerre-Gaussian beams. Phys Rev A 2016; 93: 063858. DOI: 10.1103/PhysRevA.93.063858.
  12. Kotlyar VV, Kovalev AA, Porfirev AP. Vortex Hermite-Gaussian laser beams. Opt Lett 2015; 40(5): 701-704. DOI: 10.1364/OL.40.000701.
  13. Bazhenov VY, Vasnetsov MV, Soskin MS. Laser beams with screw dislocations in their wavefront. JETP Letters 1990; 52: 429-431.
  14. Kotlyar VV, Kovalev AA, Porfirev AP. Asymmetric Gaussian optical vortex. Opt Lett 2017; 42(1): 139-142. DOI: 10.1364/OL.42.000139.
  15. Kotlyar VV, Kovalev AA, Porfirev AP, Abramochkin EG. Fractional orbital angular momentum of a Gaussian beam with an embedded off-axis optical vortex. Computer Optics 2017; 41(1): 22-29. DOI: 10.18287/2412-6179-2017-41-1-22-29.
  16. Kotlyar VV, Kovalev AA, Porfirev AP. Elliptic Gaussian optical vortices. Phys Rev A 2017; 95(5): 053805. DOI: 10.1103/PhysRevA.95.053805.
  17. Kotlyar VV, Kovalev AA. Controlling orbital angular momentum of an optical vortex by varying its ellipticity. Opt Commun 2018; 410: 202-205. DOI: 10.1016/j.optcom.2017.10.004.
  18. Prudnikov AP, Brychkov YA, Marichev OI. Integrals and series. Volume 2: Special functions. New York: Gordon and Breach; 1986. ISBN: 2-88124-097-6.

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