(43-5) 03 * << * >> * Russian * English * Content * All Issues

Sectorial perturbation of vortex beams: Shannon entropy, orbital angular momentum and topological charge

A.V. Volyar1, M.V. Bretsko1, Ya.E. Akimova1, Yu.A. Egorov1, V.V. Milyukov1

1 Physics and Technology Institute of V.I. Vernadsky Crimean Federal University,  
Academician Vernadsky 4, 295007, Simferopol, Russia

 PDF, 2977 kB

DOI: 10.18287/2412-6179-2019-43-5-723-734

Pages: 723-734.

Full text of article: Russian language.

Abstract:
Transformations of the vortex beams structure subjected to sectorial perturbation were theoretically and experimentally studied. The analysis was based on computing (measuring) the vortex spectrum that enables us to find the orbital angular momentum (OAM) and Shannon entropy (informational entropy). We have revealed that, in the general case, the number of vortices caused by an external perturbation is not related to the topological charge. For arbitrary perturbation, the topological charge remains equal to the initial topological charge of the unperturbed vortex beam. Growth of the vortex number induced by perturbations is associated with the optical uncertainty principle between the sectorial angle and the OAM. The computer simulation has shown that OAM does not depend on the number of vortices induced by perturbations. Moreover, two maxima are formed both in the positive and negative regions of the vortex spectrum. As a result, the OAM does not practically change in a wide range of perturbation angles from 0 to 90 °. However, at large perturbation angles, when the energy is almost equally redistributed between the vortex modes with opposite signs of the topological charge, the OAM rapidly decreases. At the same time, the Shannon entropy monotonically increases with growing perturbation angle. This is due to the fact that the entropy depends only on the number of vortex states caused by external perturbations.

Keywords:
diffractive optics, optical vortices, orbital angular momentum, moments of intensity.

Citation:
Volyar AV, Bretsko MV, Akimova YaE, Egorov YuA, Milyukov VV. Sectorial perturbation of vortex beams: Shannon entropy, orbital angular momentum and topological charge. Computer Optics 2019; 43(5): 723-734. DOI: 10.18287/2412-6179-2019-43-5-723-734.

Acknowledgements:
The reported study was funded by the Russian Foundation for Basic Research under RFBR grant No. 19-29-01233 and grant ВГ24, "V.I. Vernadsky Crimean Federal University Development Program in 2015–2024".

References:
  1. Yao AM, Padgett MJ. Orbital angular momentum: origins, behavior and applications. Adv Opt Photon 2011; 3: 161-204. DOI: 10.1364/AOP.3.000161.
  2. Willner AE, Huang H, Yan Y, Ren Y, Ahmed N, Xie G, Bao C, Li L, Cao Y, Zhao Z, Wang J, Lavery MPJ, Tur M, Ramachandran S, Molisch AF, Ashrafi N, Ashrafi S. Optical communications using orbital angular momentum beams. Adv Opt Photon 2015; 7: 66-106. DOI: 10.1364/AOP.7.000066.
  3. Li S, Wang J. Adaptive free-space optical communications through turbulence using self-healing Bessel beams. Sci Rep 2017; 7: 43233. DOI: 10.1038/srep43233.
  4. Vyas S, Kozawa Y, Sato S. Self-healing of tightly focused scalar and vector Bessel–Gauss beams at the focal plane. J Opt Soc Am A 2011; 28(5): 835-843. DOI: 10.1364/JOSAA.28.000837.
  5. McLaren M, Mhlanga T, Padgett MJ, Roux FS, Forbes A. Self-healing of quantum entanglement after an obstruction. Nat Commun 2014; 5: 3248. DOI: 10.1038/ncomms4248.
  6. Broky J, Siviloglou GA, Dogariu A, Christodoulides DN. Self-healing properties of optical Airy beams. Opt Express 2008; 16(17): 12880-12891. DOI: 10.1364/OE.16.012880.
  7. Aiello A, Agarwal GS, Paúr M, Stoklasa B, Hradil Z, Řeháček J, de la Hoz P, Leuchs G, Sánchez-Soto LL. Unraveling beam self-healing. Opt Express 2017; 25(16): 19147-19157. DOI: 10.1364/OE.25.019147.
  8. Ring JD, Lindberg J, Mourka A, Mazilu M, Dholakia K, Dennis MR. Auto-focusing and self-healing of Pearcey beams. Opt Express 2012; 20(17): 18955-18966. DOI: 10.1364/OE.20.018955.
  9. Mphuthi N, Boltha R, Forbes A. Are Bessel beams resilient to aberrations and turbulence? J Opt Soc Am A 2018; 35(6): 1021-1027. DOI: 10.1364/JOSAA.35.001021.
  10. Chen R, Zhang X, Zhou Y, Ming H, Wang A, Zhan Q. Detecting the topological charge of optical vortex beams using a sectorial screen. Appl Opt 2017; 56(16): 4868-4872. DOI: 10.1364/AO.56.004868.
  11. Malik M, Murugkar S, Leach J, Boyd RW. Measurement of the orbital-angular-momentum spectrum of fields with partial angular coherence using double-angular-slit interference. Phys Rev A 2012; 86: 063806. DOI: 10.1103/PhysRevA.86.063806.
  12. Franke-Arnold S, Barnett S, Yao E, Leach J, Courtial J, Padgett M. Uncertainty principle for angular position and angular momentum. New J Phys 2004; 6: 1-8. DOI: 10.1088/1367-2630/6/1/103.
  13. Yao E, Franke-Arnold S, Courtial J, Barnett S, Padgett M. Fourier relationship between angular position and optical orbital angular momentum. Opt Express 2006; 14(20): 9071-9076. DOI: 10.1364/OE.14.009071.
  14. Yu FTS. Entropy and information optics. New York: Marcel Dekker Inc; 2000: 528. ISBN: 978-0-8247-0363-9.
  15. Mandel L, Wolf E. Optical coherence and quantum optics. Cambridge: Cambridge University Press; 1995: 1192. DOI: 10.1017/CBO9781139644105.
  16. Zeng J, Lin R, Liu X, Zhao C, Cai Y. Review on partially coherent vortex beams. Frontiers of Optoelectronics 2019; 12(3): 229-248. DOI: 10.1007/s12200-019-0901-x.
  17. Wang F, Chen Y, Liu X, Cai Y, Ponarenko SA. Self-reconstruction of partially coherent light beams scattered by opaque obstacles. Opt Express 2016; 24(21): 23735-23746. DOI: 10.1364/OE.24.023735.
  18. Agarwal GS, Banerji J. Spatial coherence and information entropy in optical vortex fields. Opt Lett 2002; 27(10): 800-802. DOI: 10.1364/OL.27.000800.
  19. Kumar A, Prabhakar S, Vaity P, Singh RP. Information content of optical vortex fields. Opt Lett 2011; 36(7): 1161-1163. DOI: 10.1364/OL.36.001161.
  20. Bastiaans MJ. Uncertainty principle and informational entropy for partially coherent light. J Opt Soc Am A 1986; 3(8): 1243-1246. DOI: 10.1364/JOSAA.3.001243.
  21. Barnett SM, Pegg DT. On the Hermitian optical phase operator. In Book: Barnett SM, Vaccaro JA, eds. The quantum phase operator: a review. London: Taylor & Francis; 2007.
  22. Abramochkin E, Volostnikov V. Beam transformations and non-transformed beams. Opt Comm 1991; 83(1-2): 123-135. DOI: 10.1016/0030-4018(91)90534-K.
  23. Volyar AV, Bretsko MV, Akimova YaE, Egorov YuA. Measurement of the vortex spectrum in a vortex-beam array without cuts and gluing of the wavefront. Opt Lett 2018; 43(22): 5635-5638. DOI: 10.1364/OL.43.005635.
  24. Volyar A, Bretsko M, Akimova Y, Egorov Y. Vortex avalanche in the perturbed singular beams. J Opt Soc Am A 2019; 36(6): 1064-1071. DOI: 10.1364/JOSAA.36.001064.
  25. Berry MV. Optical vortices evolving from helicoidal integer and fractional phase steps. J Opt A 2004; 6: 259-269. DOI: 10.1088/1464-4258/6/2/018.
  26. Nye JF, Berry MV. Dislocations in wave trains. Proc R Soc A 1974; 336: 165-190. DOI: 10.1098/rspa.1974.0012.
  27. Izdebskaya Y, Shvedov V, Volyar A. Focusing of wedge-generated higher-order optical vortices. Opt Lett 2005; 30(19): 2530-2532. DOI: 10.1364/OL.30.002530.
  28. Volyar AV, Zhilaitis VZ, Shvedov VG. Optical eddies in small-mode fibers: II. The spin-orbit interaction. Optika i Spektroskopiya 1999; 86(4): 664-670.
  29. Alexeyev CN, Egorov YuA, Volyar AV. Mutual transformations of fractional-order and integer-order optical vortices. Phys Rev A 2017; 96: 063807. DOI: 10.1103/PhysRevA.96.063807.
  30. Khonina SN, Kotlyar VV, Soifer VA, Jefimovs K, Turunen J. Generation and selection of laser beams represented by a superposition of two angular harmonics. J Mod Opt 2004; 51(5): 761-773. DOI: 10.1080/09500340408235551.
  31. Wang Z, Zhang N, Yuan X-C. High-volume optical vortex multiplexing and de-multiplexing for free-space optical communication. Opt Express 2011; 19(2): 482-492. DOI: 10.1364/OE.19.000482.
  32. 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.
  33. Volyar A, Bretsko M, Akimova Y, Egorov Y. Shaping and processing the vortex spectra of singular beams with anomalous orbital angular momentum. Computer Optics 2019; 43(4): 517-527. DOI: 10.18287/2412-6179-2019-43-4-517-527.
  34. Gonzalez RC, Woods RE. Digital image processing. New Jersey: Prentice Hall Upper Saddle River; 2002.

 

© 2009, IPSI RAS
Россия, 443001, Самара, ул. Молодогвардейская, 151; электронная почта: ko@smr.ru ; тел: +7 (846) 242-41-24 (ответственный секретарь), +7 (846) 332-56-22 (технический редактор), факс: +7 (846) 332-56-20