Cross-phase modulation instability under delayed response and walk-off effects in the anomalous regime of dispersion

  • Askery CanabarroEmail author
  • Beethoven Santos
  • Bertulio de Lima Bernardo
Regular Article


The dynamics of the modulation instability induced by cross phase modulation is studied by considering the influence of the walk-off and noninstantaneous response effects for two copropagating optical fields travelling in the anomalous regime of dispersion. To do so, we make use of extensions of the nonlinear Schrdinger equation jointly with the Debye model for polarization, which is shown to be effective and simplified when compared to the implementation of other methods for the noninstantaneous nonlinear response. In analyzing the sideband formation, two bands are observed with different behaviors with respect to the way the maximum gain and the respective frequency vary with increasing phase mismatch. Further, we also show the manner in which the maximum gain as well as its corresponding frequency scale with the delay parameter τ is substantially different from the cases of both fields experiencing the normal group-velocity dispersion regime and the case of mixed regimes. These facts may give rise to many new possibilities for the evolution of the modulated wave when compared to the nonanomalous cases studied in the literature.

Graphical abstract


Optical Phenomena and Photonics 


  1. 1.
    T.B. Benjamin, J.E. Feir, J. Fluid Mech. 27, 417430 (1967)CrossRefGoogle Scholar
  2. 2.
    G.P. Agrawal, J. Opt. Soc. Am. B 28, A1 (2011)ADSCrossRefGoogle Scholar
  3. 3.
    P.A.E.M. Janssen, J. Phys. Oceanogr. 33, 863 (2003)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    K. Dysthe, H.E. Krogstad, P. Müller, Ann. Rev. Fluid Mech. 40, 287 (2008)ADSCrossRefGoogle Scholar
  5. 5.
    M. Onorato, A.R. Osborne, M. Serio, Phys. Rev. Lett. 96, 014503 (2006)ADSCrossRefGoogle Scholar
  6. 6.
    M. Onorato, A.R. Osborne, M. Serio, S. Bertone, Phys. Rev. Lett. 86, 5831 (2001)ADSCrossRefGoogle Scholar
  7. 7.
    D.R. Solli, C. Ropers, P. Koonath, B. Jalali, Nature 450, 1054 (2007)ADSCrossRefGoogle Scholar
  8. 8.
    N.V. Priya, M. Senthilvelan, Wave Motion 54, 125 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    M. Stepić, A. Maluckov, M. Stojanović, F. Chen, D. Kip, Phys. Rev. A 78, 043819 (2008)ADSCrossRefGoogle Scholar
  10. 10.
    P.J. Everitt, M.A. Sooriyabandara, M. Guasoni, P.B. Wigley, C.H. Wei, G.D. McDonald, K.S. Hardman, P. Manju, J.D. Close, C.C.N. Kuhn, S.S. Szigeti, Y.S. Kivshar, N.P. Robins, Phys. Rev. A 96, 041601 (2017)ADSCrossRefGoogle Scholar
  11. 11.
    A.G. Litvak, Zh. Eksp, Teor. Fiz. 57, 629 (1969)Google Scholar
  12. 12.
    M. Wu, B.A. Kalinikos, Phys. Rev. Lett. 101, 027206 (2008)ADSCrossRefGoogle Scholar
  13. 13.
    V. Zakharov, L. Ostrovsky, Physica D 238, (5402009)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    G.P. Agrawal, Phys. Rev. Lett. 59, 880 (1987)ADSCrossRefGoogle Scholar
  15. 15.
    Y. Kivshar, G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Elsevier Science, 2003)Google Scholar
  16. 16.
    G. Biondini, D. Mantzavinos, Phys. Rev. Lett. 116, 043902 (2016)ADSCrossRefGoogle Scholar
  17. 17.
    L.F. Mollenauer, R.H. Stolen, J.P. Gordon, Phys. Rev. Lett. 45, 1095 (1980)ADSCrossRefGoogle Scholar
  18. 18.
    G.P. Agrawal, P.L. Baldeck, R.R. Alfano, Phys. Rev. A 39, 3406 (1989)ADSCrossRefGoogle Scholar
  19. 19.
    K. Chow, K. Wong, K. Lam, Phys. Lett. A 372, 4596 (2008)ADSCrossRefGoogle Scholar
  20. 20.
    L. Zhang, Y. Xiang, X. Dai, S. Wen, J. Opt. Soc. Am. B 31, 3029 (2014)ADSCrossRefGoogle Scholar
  21. 21.
    K. Nithyanandan, R.V.J. Raja, K. Porsezian, B. Kalithasan, Phys. Rev. A 86, 023827 (2012)ADSCrossRefGoogle Scholar
  22. 22.
    A. Biswas, K.R. Khan, M.F. Mahmood, M. Belic, Optik 125, 3299 (2014)ADSCrossRefGoogle Scholar
  23. 23.
    A.S. Ali, K. Nithyanandan, K. Porsezian, Phys. Lett. A 379, 223 (2015)ADSCrossRefGoogle Scholar
  24. 24.
    Y. Xiang, X. Dai, S. Wen, D. Fan, J. Opt. Soc. Am. B 28, 908 (2011)ADSCrossRefGoogle Scholar
  25. 25.
    W. Zhou, W. Su, X. Cheng, Y. Xiang, X. Dai, S. Wen, Opt. Commun. 282, 1440 (2009)ADSCrossRefGoogle Scholar
  26. 26.
    X. Zhong, K. Cheng, K.S. Chiang, J. Opt. Soc. Am. B 31, 1484 (2014)ADSCrossRefGoogle Scholar
  27. 27.
    X. Zhong, A. Xiang, Opt. Fiber Tech. 13, 271 (2007)ADSCrossRefGoogle Scholar
  28. 28.
    G.L. da Silva, I. Gleria, M.L. Lyra, A.S.B. Sombra, J. Opt. Soc. Am. B 26, 183 (2009)ADSCrossRefGoogle Scholar
  29. 29.
    T. Tanemura, K. Kikuchi, J. Opt. Soc. Am. B 20, 2502 (2003)ADSCrossRefGoogle Scholar
  30. 30.
    E. Kengne, S.T. Chui, W.M. Liu, Phys. Rev. E 74, 036614 (2006)ADSCrossRefGoogle Scholar
  31. 31.
    R.V.J. Raja, A. Husakou, J. Hermann, K. Porsezian, J. Opt. Soc. Am. B 27, 1763 (2010)ADSCrossRefGoogle Scholar
  32. 32.
    W. Królikowski, O. Bang, N.I. Nikolov, D. Neshev, J. Wyller, J.J. Rasmussen, D. Edmundson, J. Opt. B: Quantum Semiclassical Opt. 6, S288 (2004)ADSCrossRefGoogle Scholar
  33. 33.
    Q. Lin, G.P. Agrawal, IEEE J. Quantum Electron. 40, 958 (2004)ADSCrossRefGoogle Scholar
  34. 34.
    J.E. Rothenberg, Phys. Rev. A 42, 682 (1990)ADSCrossRefGoogle Scholar
  35. 35.
    A. Canabarro, B. Santos, B. de Lima Bernardo, A.L. Moura, W.C. Soares, E. de Lima, I. Gléria, M.L. Lyra, Phys. Rev. A 93, 023834 (2016)ADSCrossRefGoogle Scholar
  36. 36.
    G. da Silva, A. Canabarro, B. de Lima Bernardo, Ann. Phys. 363, 476 (2015)ADSCrossRefGoogle Scholar
  37. 37.
    A.A. Canabarro, B. Santos, I. Gleria, M.L. Lyra, A.S.B. Sombra, J. Opt. Soc. Am. B 27, 1878 (2010)ADSCrossRefGoogle Scholar
  38. 38.
    G.L. da Silva, T.P. Lobo, A.A. Canabarro, J. Opt. Soc. Am. B 31, 2012 (2014)ADSCrossRefGoogle Scholar
  39. 39.
    S.E. Harris, Y. Yamamoto, Phys. Rev. Lett. 81, 3611 (1998)ADSCrossRefGoogle Scholar
  40. 40.
    H. Schmidt, A. Imamoglu, Opt. Lett. 21, 1936 (1996)ADSCrossRefGoogle Scholar
  41. 41.
    I. Fushman, J. Vučković, Opt. Express 15, 5559 (2007)ADSCrossRefGoogle Scholar
  42. 42.
    M. Hallaji, A. Feizpour, G. Dmochowski, J. Sinclair, A.M. Steinberg, How a single photon can act like many photons, in 2015 European Conference on Lasers and Electro-Optics – European Quantum Electronics Conference (Optical Society of America, 2015)Google Scholar
  43. 43.
    K. Nithyanandan, K. Porsezian, Opt. Commun. 303, 46 (2013)ADSCrossRefGoogle Scholar
  44. 44.
    C.-S. Chou, M.-F. Shih, J. Opt. A: Pure Appl. Opt. 11, 105204 (2009)ADSCrossRefGoogle Scholar
  45. 45.
    C. Conti, M.A. Schmidt, P.S.J. Russell, F. Biancalana, Phys. Rev. Lett. 105, 263902 (2010)ADSCrossRefGoogle Scholar
  46. 46.
    N.N. Akhmediev, M.J. Lederer, B. Luther-Davies, Phys. Rev. E 57, 3664 (1998)ADSCrossRefGoogle Scholar
  47. 47.
    G. Carbou, B. Hanouzet, Commun. Math. Sci. 4, 331 (2006)MathSciNetCrossRefGoogle Scholar
  48. 48.
    C. Cambournac, H. Maillotte, E. Lantz, J.M. Dudley, M. Chauvet, J. Opt. Soc. Am. B 19, 574 (2002)ADSCrossRefGoogle Scholar
  49. 49.
    R.W. Ziolkowski, IEEE Trans. Antennas Propag. 45, 375 (1997)ADSCrossRefGoogle Scholar
  50. 50.
    R.W. Ziolkowski, J.B. Judkins, J. Opt. Soc. Am. B 10, 186 (1993)ADSCrossRefGoogle Scholar
  51. 51.
    M.J. Potasek, Opt. Lett. 12, 921 (1987)ADSCrossRefGoogle Scholar
  52. 52.
    X. Liu, J.W. Haus, S.M. Shahriar, Opt. Commun. 281, 2907 (2008)ADSCrossRefGoogle Scholar
  53. 53.
    S. Trillo, S. Wabnitz, G.I. Stegeman, E.M. Wright, J. Opt. Soc. Am. B 6, 889 (1989)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences / Società Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.International Institute of Physics, Federal University of Rio Grande do NorteNatalBrazil
  2. 2.Grupo de Física da Matéria Condensada, Núcleo de Ciências Exatas, Campus Arapiraca, Universidade Federal de AlagoasArapiracaBrazil
  3. 3.Observatório NacionalRio de JaneiroBrazil
  4. 4.Departamento de Física, Universidade Federal da ParaíbaJoão PessoaBrazil

Personalised recommendations