Nonlinear Dynamics

, Volume 93, Issue 4, pp 2159–2168 | Cite as

Generation of ring-shaped optical vortices in dissipative media by inhomogeneous effective diffusion

  • Shiquan Lai
  • Huishan Li
  • Yunli Qui
  • Xing Zhu
  • Dumitru Mihalache
  • Boris A. Malomed
  • Yingji HeEmail author
Original Paper


By means of systematic simulations, we demonstrate generation of a variety of ring-shaped optical vortices (OVs) from a two-dimensional input with embedded vorticity, in a dissipative medium modeled by the cubic–quintic complex Ginzburg–Landau equation with an inhomogeneous effective diffusion (spatial filtering) term, which is anisotropic in the transverse plane and periodically modulated in the longitudinal direction. We show the generation of stable square- and gear-shaped OVs, as well as tilted oval-shaped vortex rings, and string-shaped bound states built of a central fundamental soliton and two vortex satellites, or of three fundamental solitons. Their shape can be adjusted by tuning the strength and modulation period of the inhomogeneous diffusion. Stability domains of the generated OVs are identified by varying the vorticity of the input and parameters of the inhomogeneous diffusion. The results suggest a method to generate new types of ring-shaped OVs with applications to the work with structured light.


Inhomogeneous effective diffusion Ginzburg–Landau equation Optical vortices 



This work was supported by the National Natural Science Foundations of China (Grant Nos. 11174061, 61675001, and 11774068), the Guangdong Province Nature Foundation of China (Grant No. 2017A030311025), and the Guangdong Province Education Department Foundation of China (Grant No. 2014KZDXM059). We declare that we do not have any conflict of interest in connection with the present work.


  1. 1.
    Kivshar, Y.S., Agrawal, G.P.: Optical Solitons: From Fibers to Photonic Cristals. Academic Press, San Diego (2003)Google Scholar
  2. 2.
    Firth, W.J.: In: Vorontsov, M.A., Miller, W.B. (eds.) Self-Organization in Optical Systems and Applications in Information Technology. Springer, Berlin (1995)Google Scholar
  3. 3.
    Malomed, B.A., Mihalache, D., Wise, F., Torner, L.: Spatiotemporal optical solitons. J. Opt. B: Quantum Semiclass. Opt. 7, R53–R72 (2005)CrossRefGoogle Scholar
  4. 4.
    Mihalache, D.: Linear and nonlinear light bullets: recent theoretical studies. Rom. J. Phys. 57, 352–371 (2012)Google Scholar
  5. 5.
    Bagnato, V.S., Frantzeskakis, D.J., Kevrekidis, P.G., Malomed, B.A., Mihalache, D.: Bose–Einstein condensation: twenty years after. Rom. Rep. Phys. 67, 5–50 (2015)Google Scholar
  6. 6.
    Malomed, B., Torner, L., Wise, F., Mihalache, D.: On multidimensional solitons and their legacy in contemporary atomic, molecular and optical physics. J. Phys. B: At. Mol. Opt. Phys. 49, 170502 (2016)CrossRefGoogle Scholar
  7. 7.
    Malomed, B.A.: Multidimensional solitons: well-established results and novel findings. Eur. Phys. J. Spec. Top. 225, 2507–2532 (2016)CrossRefGoogle Scholar
  8. 8.
    Rosanov, N.N., Fedorov, S.V., Shatsev, A.N.: In: Akhmediev, N., Ankiewicz, A. (eds.) Dissipative Solitons: From Optics to Biology and Medicine. Lecture Notes in Physics, vol. 751. Springer, Berlin (2008)Google Scholar
  9. 9.
    Kuszelewicz, R., Barbay, S., Tissoni, G., Almuneau, G.: Editorial on dissipative optical solitons. Eur. Phys. J. D 59(1), 1–2 (2010)CrossRefGoogle Scholar
  10. 10.
    Firth, W.J., Scroggie, A.J.: Optical bullet holes: robust controllable localized states of a nonlinear cavity. Phys. Rev. Lett. 76(10), 1623–1626 (1996)CrossRefGoogle Scholar
  11. 11.
    Chen, Z., Mccarthy, K.: Spatial soliton pixels from partially incoherent light. Opt. Lett. 27(22), 2019–2021 (2002)CrossRefGoogle Scholar
  12. 12.
    Fleischer, J.W., Segev, M., Efremidis, N.K., Christodoulides, D.N.: Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices. Nature 422(6928), 147–150 (2003)CrossRefGoogle Scholar
  13. 13.
    Kartashov, Y.V., Egorov, A.A., Torner, L., Christodoulides, D.N.: Stable soliton complexes in two-dimensional photonic lattices. Opt. Lett. 29(16), 1918–1920 (2004)CrossRefGoogle Scholar
  14. 14.
    Rosanov, N.N.: Spatial Hysteresis and Optical Patterns. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  15. 15.
    Malomed, B.A.: Complex Ginzburg-Landau equation. In: Scott, A. (ed.) Encyclopedia of Nonlinear Science, pp. 157–160. Routledge, New York (2005)Google Scholar
  16. 16.
    Mandel, P., Tlidi, M.: Transverse dynamics in cavity nonlinear optics. J. Opt. B 6, R60–R75 (2004)CrossRefGoogle Scholar
  17. 17.
    Akhmediev, N.N., Afanasjev, V.V., Soto-Crespo, J.M.: Singularities and special soliton solutions of the cubic–quintic complex Ginzburg–Landau equation. Phys. Rev. E 53, 1190–1201 (1996)CrossRefGoogle Scholar
  18. 18.
    Mihalache, D., Mazilu, D., Lederer, F., Kartashov, Y.V., Crasovan, L.C., Torner, L., Malomed, B.A.: Stable vortex tori in the three-dimensional cubic–quintic Ginzburg–Landau equation. Phys. Rev. Lett. 97, 073904 (2006)CrossRefGoogle Scholar
  19. 19.
    Leblond, H., Komarov, A., Salhi, M., Haboucha, A., Sanchez, F.: Bound states of three localized states of the cubic–quintic CGL equation. J. Opt. A 8, 319–326 (2006)CrossRefGoogle Scholar
  20. 20.
    Mihalache, D., Mazilu, D., Lederer, F., Leblond, H., Malomed, B.A.: Stability of dissipative optical solitons in the three-dimensional cubic–quintic Ginzburg–Landau equation. Phys. Rev. A 75, 033811 (2007)CrossRefGoogle Scholar
  21. 21.
    Renninger, W.H., Chong, A., Wise, F.W.: Dissipative solitons in normal-dispersion fiber lasers. Phys. Rev. A 77, 023814 (2008)CrossRefGoogle Scholar
  22. 22.
    Taki, M., Akhmediev, N., Wabnitz, S., Chang, W.: Influence of external phase and gain-loss modulation on bound solitons in laser systems. J. Opt. Soc. Am. B 26, 2204–2210 (2009)CrossRefGoogle Scholar
  23. 23.
    Aranson, I.S., Kramer, L.: The world of the complex Ginzburg–Landau equation. Rev. Mod. Phys. 74, 99–143 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Akhmediev, N., Sotocrespo, J.M., Town, G.: Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: complex Ginzburg–Landau equation approach. Phys. Rev. E 63, 056602 (2001)CrossRefGoogle Scholar
  25. 25.
    Crasovan, L.C., Malomed, B.A., Mihalache, D.: Stable vortex solitons in the two-dimensional Ginzburg–Landau equation. Phys. Rev. E 63, 016605 (2001)CrossRefzbMATHGoogle Scholar
  26. 26.
    He, Y.J., Malomed, B.A., Ye, F.W., Hu, B.B.: Dynamics of dissipative spatial solitons over a sharp potential. J. Opt. Soc. Am. B 27, 1139–1142 (2010)CrossRefGoogle Scholar
  27. 27.
    Grelu, P., Akhmediev, N.: Dissipative solitons for mode-locked lasers. Nat. Photon. 6, 84–92 (2012)CrossRefGoogle Scholar
  28. 28.
    Fernandez-Oto, C., Valcárcel, G.J.D., Tlidi, M., Panajotov, K., Staliunas, K.: Phase-bistable patterns and cavity solitons induced by spatially periodic injection into vertical-cavity surface-emitting lasers. Phys. Rev. A 89, 055802 (2014)CrossRefGoogle Scholar
  29. 29.
    Malomed, B.A.: Spatial solitons supported by localized gain. J. Opt. Soc. Am. B 31, 2460–2475 (2014)CrossRefGoogle Scholar
  30. 30.
    Tlidi, M., Staliunas, K., Panajotov, K., Vladimirov, A.G., Clerc, M.G.: Introduction: localized structures in dissipative media—from optics to plant ecology. Phil. Trans. R. Soc. A 372, 20140101 (2014)CrossRefGoogle Scholar
  31. 31.
    Rosanov, N.N., Sochilin, G.B., Vinokurova, V.D., Vysotina, N.V.: Spatial and temporal structures in cavities with oscillating boundaries. Phil. Trans. R. Soc. A 372, 20140012 (2014)CrossRefGoogle Scholar
  32. 32.
    Mihalache, D., Mazilu, D., Skarka, V., Malomed, B.A., Leblond, H., Aleksi, N.B., Lederer, F.: Stable topological modes in two-dimensional Ginzburg–Landau models with trapping potentials. Phys. Rev. A 82, 023813 (2010)CrossRefGoogle Scholar
  33. 33.
    Skarka, V., Aleksić, N.B., Leblond, H., Malomed, B.A., Mihalache, D.: Varieties of stable vortical solitons in Ginzburg–Landau media with radially inhomogeneous losses. Phys. Rev. Lett. 105, 213901 (2010)CrossRefGoogle Scholar
  34. 34.
    Skarka, V., Aleksić, N.B., Lekić, M., Aleksić, B.N., Malomed, B.A., Mihalache, D., Leblond, H.: Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking. Phys. Rev. A 90, 023845 (2014)CrossRefGoogle Scholar
  35. 35.
    Liu, B., He, X.-D., Li, S.-J.: Continuous emission of fundamental solitons from vortices in dissipative media by a radial-azimuthal potential. Opt. Express 21(5), 5561–5566 (2013)CrossRefGoogle Scholar
  36. 36.
    Liu, B., Liu, Y.F., He, X.D.: Impact of phase on collision between vortex solitons in three-dimensional cubic–quintic complex Ginzburg–Landau equation. Opt. Express 22(21), 26203–26211 (2014)CrossRefGoogle Scholar
  37. 37.
    Kalasnikov, V.L., Sorokin, E.: Dissipative raman solitons. Opt. Express 22(24), 30118–30126 (2014)CrossRefGoogle Scholar
  38. 38.
    Song, Y.F., Zhang, H., Zhao, L.M., Shen, D.Y., Tang, D.Y.: Coexistence and interaction of vector and bound vector solitons in a dispersion-managed fiber laser mode locked by graphene. Opt. Express 24(2), 1814–1822 (2016)CrossRefGoogle Scholar
  39. 39.
    Mihalache, D.: Localized optical structures: an overview of recent theoretical and experimental developments. Proc. Rom. Acad. A 16(1), 62–69 (2015)MathSciNetGoogle Scholar
  40. 40.
    Mihalache, D.: Multidimensional localized structures in optical and matter-wave media: a topical survey of recent literature. Rom. Rep. Phys. 69(1), 403 (2017)Google Scholar
  41. 41.
    Skarka, V., Aleksić, N.B., Krolikowski, W., Christodoulides, D.N., Rakotoarimalala, S., Aleksić, B.N., Belić, M.: Self-structuring of stable dissipative breathing vortex solitons in a colloidal nanosuspension. Opt. Express 25(9), 10090–10102 (2017)CrossRefGoogle Scholar
  42. 42.
    Swartzlander, G.A., Law, C.T.: Optical vortex solitons observed in Kerr nonlinear media. Phys. Rev. Lett. 69(17), 2503–2506 (1992)CrossRefGoogle Scholar
  43. 43.
    Lutherdavies, B., Christou, J., Tikhonenko, V., Kivshar, Y.S.: Optical vortex solitons: experiment versus theory. J. Opt. Soc. Am. B 14(11), 3045–3053 (1997)CrossRefGoogle Scholar
  44. 44.
    Desyatnikov, A.S., Torner, L., Kivshar, Y.S.: Optical vortices and vortex solitons. Progr. Opt. 47, 291–391 (2005)CrossRefGoogle Scholar
  45. 45.
    Tikhonenko, V., Akhmediev, N.N.: Excitation of vortex solitons in a Gaussian beam configuration. Opt. Commun. 126, 108–112 (1996)CrossRefGoogle Scholar
  46. 46.
    Carlsson, A.H., Dan, A., Ostrovskaya, E.A., Malmberg, J.N., Lisak, M., Alexander, T.J., Kivshar, Y.S.: Linear and nonlinear waveguides induced by optical vortex solitons. Opt. Lett. 25, 660–662 (2000)CrossRefGoogle Scholar
  47. 47.
    Kivshar, Y.S., Luther-Davies, B.: Dark optical solitons: physics and applications. Phys. Rep. 298, 81–197 (1998)CrossRefGoogle Scholar
  48. 48.
    Andrews, D.: Structured light and its applications: an introduction to phase-structured beams and nanoscale optical forces. Academic Press, Cambridge (2008)Google Scholar
  49. 49.
    Adhikari, S.K.: Stable spatial and spatiotemporal optical soliton in the core of an optical vortex. Phys. Rev. E 92, 042926 (2015)CrossRefGoogle Scholar
  50. 50.
    Carlsson, A.H., Ostrovskaya, E., Salgueiro, J.R., Kivshar, Y.: Second-harmonic generation in vortex-induced waveguides. Opt. Lett. 29, 593–595 (2004)CrossRefGoogle Scholar
  51. 51.
    Yao, A.M., Padgett, M.J.: Orbital angular momentum: origins, behavior and applications. Adv. Opt. Photon. 3(2), 161–204 (2011)CrossRefGoogle Scholar
  52. 52.
    Law, C.T., Zhang, X., Swartzlander, G.A.: Waveguiding properties of optical vortex solitons. Opt. Lett. 25(1), 55–57 (2000)CrossRefGoogle Scholar
  53. 53.
    Ashkin, A., Dziedzic, J.M., Bjorkholm, J.E., Chu, S.: Observation of a single-beam gradient force optical trap for dielectric particles. Opt. Lett. 11, 288–290 (1986)CrossRefGoogle Scholar
  54. 54.
    Kivshar, Y.S., Christou, J., Tikhonenko, V., Luther-Davies, B., Pismen, L.M.: Dynamics of optical vortex solitons. Opt. Commun. 152, 198–206 (1998)CrossRefGoogle Scholar
  55. 55.
    Velchev, I., Dreischuh, A., Neshev, D., Dinev, S.: Interactions of optical vortex solitons superimposed on different background beams. Opt. Commun. 130, 385–392 (1996)CrossRefGoogle Scholar
  56. 56.
    Rozas, D., Swartzlander, G.A.: Observed rotational enhancement of nonlinear optical vortices. Opt. Lett. 25, 126–128 (2000)CrossRefGoogle Scholar
  57. 57.
    Rozas, D., Law, C.T., Swartzlander, G.A.: Propagation dynamics of optical vortices. J. Opt. Soc. Am. B 14, 3054–3065 (1997)CrossRefGoogle Scholar
  58. 58.
    Neshev, D., Dreischuh, A., Assa, M., Dinev, S.: Motion control of ensembles of ordered optical vortices generated on finite extent background. Opt. Commun. 151, 413–421 (1998)CrossRefGoogle Scholar
  59. 59.
    Huang, C., Ye, F., Malomed, B.A., Kartashov, Y.V., Chen, X.: Solitary vortices supported by localized parametric gain. Opt. Lett. 38, 2177–2180 (2013)CrossRefGoogle Scholar
  60. 60.
    Zeng, J., Malomed, B.A.: Localized dark solitons and vortices in defocusing media with spatially inhomogeneous nonlinearity. Phys. Rev. E 95, 052214 (2017)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Reyna, A.S., de Araújo, C.B.: Guiding and confinement of light induced by optical vortex solitons in a cubic–quintic medium. Opt. Lett. 41, 191–194 (2016)CrossRefGoogle Scholar
  62. 62.
    Hu, B., Ye, F., Torner, L., Kartashov, Y.V.: Twin-vortex solitons in nonlocal nonlinear media. Opt. Lett. 35, 628–630 (2010)CrossRefGoogle Scholar
  63. 63.
    Porras, M.A., Ramos, F.: Quasi-ideal dynamics of vortex solitons embedded in flattop nonlinear Bessel beams. Opt. Lett. 42, 3275–3278 (2017)CrossRefGoogle Scholar
  64. 64.
    Lai, X.-J., Cai, X.-O., Zhang, J.-F.: Compression and stretching of ring-vortex solitons in a bulk nonlinear medium. Chin. Phys. B 24, 070503 (2015)CrossRefGoogle Scholar
  65. 65.
    Huang, C., Dong, L.: Stable vortex solitons in a ring-shaped partially-PT-symmetric potential. Opt. Lett. 41, 5194–5197 (2016)CrossRefGoogle Scholar
  66. 66.
    Lou, J., Cheng, M., Lim, T.T.: Evolution of an elliptic vortex ring in a viscous fluid. Phys. Fluids 28, 037104 (2016)CrossRefGoogle Scholar
  67. 67.
    Veretenov, N.A., Rosanov, N.N., Fedorov, S.V.: Motion of complexes of 3D-laser solitons. Opt. Quant. Electron. 40, 253–262 (2008)CrossRefGoogle Scholar
  68. 68.
    Veretenov, N.A., Fedorov, S.V., Rosanov, N.N.: Topological vortex and knotted dissipative optical 3D solitons generated by 2D vortex solitons. Phys. Rev. Lett. 119, 263901 (2017)CrossRefGoogle Scholar
  69. 69.
    Skupin, S., Bergé, L., Peschel, U., Lederer, F., Méjean, G., Yu, J., Kasparian, J., Salmon, E., Wolf, J.P., Rodriguez, M., Wöste, L., Bourayou, R., Sauerbrey, R.: Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters. Phys. Rev. E 70, 046602 (2004)CrossRefGoogle Scholar
  70. 70.
    Coullet, P., Gil, L., Rocca, F.: Optical vortices. Opt. Commun. 73, 403–408 (1989)CrossRefGoogle Scholar
  71. 71.
    Lega, J., Moloney, J.V., Newell, A.C.: Swift–Hohenberg equation for lasers. Phys. Rev. Lett. 73, 2978–2981 (1994)CrossRefGoogle Scholar
  72. 72.
    Hochheiser, D., Moloney, J.V., Lega, J.: Controlling optical turbulence. Phys. Rev. A 55, R4011–R4014 (1997)CrossRefGoogle Scholar
  73. 73.
    Askitopoulos, A., Ohadi, H., Hatzopoulos, Z., Savvidis, P.G., Kavokin, A.V., Lagoudakis, P.G.: Polariton condensation in an optically induced two-dimensional potential. Phys. Rev. B 88, 041308 (2013)CrossRefGoogle Scholar
  74. 74.
    Ardizzone, V., Lewandowski, P., Luk, M.H., Tse, Y.C., Kwong, N.H., Lücke, A., Abbarchi, M., Baudin, E., Galopin, E., Bloch, J., Lemaitre, A., Leung, P.T., Roussignol, P., Binder, R., Tignon, J., Schumacher, S.: Formation and control of tturing patterns in a coherent quantum fluid. Sci. Rep. 3, 3016 (2013)CrossRefGoogle Scholar
  75. 75.
    Schachenmayer, J., Genes, C., Tignone, E., Pupillo, G.: Cavity-enhanced transport of excitons. Phys. Rev. Lett. 114, 196403 (2015)CrossRefGoogle Scholar
  76. 76.
    Shahnazaryan, V., Kyriienko, O., Shelykh, I.: Adiabatic preparation of a cold exciton condensate. Phys. Rev. B 91, 085302 (2014)CrossRefGoogle Scholar
  77. 77.
    Bobrovska, N., Matuszewski, M.: Adiabatic approximation and fluctuations in exciton-polariton condensates. Phys. Rev. B 92, 035311 (2015)CrossRefGoogle Scholar
  78. 78.
    Li, H., Lai, S., Qui, Y., Zhu, X., Xie, J., Mihalache, D., He, Y.: Stable dissipative optical vortex clusters by inhomogeneous effective diffusion. Opt. Express 25, 27948–27967 (2017)CrossRefGoogle Scholar
  79. 79.
    Malomed, B.A.: Soliton Management in Periodic Systems. Springer, New York (2006)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Shiquan Lai
    • 1
  • Huishan Li
    • 1
  • Yunli Qui
    • 1
  • Xing Zhu
    • 2
  • Dumitru Mihalache
    • 3
  • Boris A. Malomed
    • 4
  • Yingji He
    • 1
    Email author
  1. 1.School of Photoelectric EngineeringGuangdong Polytechnic Normal UniversityGuangzhouChina
  2. 2.Department of Physics and Information EngineeringGuangdong University of EducationGuangzhouChina
  3. 3.Horia Hulubei National Institute for Physics and Nuclear EngineeringBucharest, MagureleRomania
  4. 4.Department of Physical Electronics, School of Electrical Engineering, Faculty of EngineeringTel Aviv UniversityTel AvivIsrael

Personalised recommendations