Semiconductor mode-locked laser with external feedback: emergence of multi-frequency pulse trains with an increasing number of modes

  • Kathy Lüdge
  • Lina Jaurigue
  • Benjamin Lingnau
  • Soizic Terrien
  • Bernd KrauskopfEmail author
Regular Article
Part of the following topical collections:
  1. Topical issue: Non-Linear and Complex Dynamics in Semiconductors and Related Materials


We investigate a mode-locked laser with optical feedback and a spectral filter that limits the number of lasing modes. For up to around ten modes the laser effectively behaves as a single-mode laser: it may generate Q-switched pulses but does not mode lock. When the number of modes is increased we observe the emergence of multi-frequency pulsing associated with invariant tori and multistability. We model this system with a delay differential equation system for the electric field, the gain and the absorption, and present a bifurcation analysis of the route to this type of pulsing. For increasing spectral filter width we present bifurcation diagrams in the plane of delay time and strength of the external feedback loop. The associated dynamics are represented by means of 2D period-roundtrip diagrams and 2D spectral plots. We illuminate the emergence of dynamics on different tori and the corresponding frequencies involved.

Graphical abstract


  1. 1.
    T. Udem, R. Holzwarth, T.W. Hänsch, Nature 416, 233 (2002) ADSCrossRefGoogle Scholar
  2. 2.
    U. Keller, Nature 424, 831 (2003) ADSCrossRefGoogle Scholar
  3. 3.
    G.J. Spühler, P.S. Golding, L. Krainer, I.J. Kilburn, P.A. Crosby, M. Brownell, K.J. Weingarten, R. Paschotta, M. Haiml, R. Grange, U. Keller, Electr. Lett. 39, 778 (2003) CrossRefGoogle Scholar
  4. 4.
    K. Lüdge, Nonlinear Laser Dynamics – From Quantum Dots to Cryptography (Wiley-, Weinheim, 2012) Google Scholar
  5. 5.
    D.J. Derickson, R.J. Helkey, A. Mar, J.R. Karin, J.G. Wasserbauer, J.E. Bowers, IEEE J. Quantum Electr. 28, 2186 (1992) ADSCrossRefGoogle Scholar
  6. 6.
    O. Solgaard, K.Y. Lau, IEEE Photon. Technol. Lett. 5, 1264 (1993) ADSCrossRefGoogle Scholar
  7. 7.
    E.A. Avrutin, J.H. Marsh, E.L. Portnoi, IEE Proc. Optoelectr. 147, 251 (2000) CrossRefGoogle Scholar
  8. 8.
    E.U. Rafailov, M.A. Cataluna, W. Sibbett, Nat. Photonics 1, 395 (2007) ADSCrossRefGoogle Scholar
  9. 9.
    D. Arsenijević, M. Kleinert, D. Bimberg, Appl. Phys. Lett. 103, 231101 (2013) ADSCrossRefGoogle Scholar
  10. 10.
    O. Nikiforov, L.C. Jaurigue, L. Drzewietzki, K. Lüdge, S. Breuer, Opt. Express 24, 14301 (2016) ADSCrossRefGoogle Scholar
  11. 11.
    D.J. Derickson, P.A. Morton, J.E. Bowers, R.L. Thornton, Appl. Phys. Lett. 59, 3372 (1991) ADSCrossRefGoogle Scholar
  12. 12.
    M.G. Thompson, C. Marinelli, K.T. Tan, K.A. Williams, R.V. Penty, I.H. White, I.N. Kaiander, R.L. Sellin, D. Bimberg, D.J. Kang, M.G. Blamire, F. Visinka, S. Jochum, S. Hansmann, Electr. Lett. 39, 1121 (2003) CrossRefGoogle Scholar
  13. 13.
    D. Bimberg, G. Fiol, M. Kuntz, C. Meuer, M. Lämmlin, N.N. Ledentsov, A.R. Kovsh, Phys. Stat. Sol. A 203, 3523 (2006) ADSCrossRefGoogle Scholar
  14. 14.
    R.M. Arkhipov, A.S. Pimenov, M. Radziunas, D. Rachinskii, A.G. Vladimirov, D. Arsenijević, H. Schmeckebier, D. Bimberg, IEEE J. Quantum Electr. 19, 1100208 (2013) CrossRefGoogle Scholar
  15. 15.
    Z. Ahmed, H.F. Liu, D. Novak, Y. Ogawa, M.D. Pelusi, D.Y. Kim, IEEE Photon. Technol. Lett. 8, 37 (1996) ADSCrossRefGoogle Scholar
  16. 16.
    N. Rebrova, T. Habruseva, G. Huyet, S.P. Hegarty, Appl. Phys. Lett. 97, 1 (2010) CrossRefGoogle Scholar
  17. 17.
    T. Habruseva, G. Huyet, S.P. Hegarty, IEEE J. Sel. Top. Quantum Electron. 17, 1272 (2011) ADSCrossRefGoogle Scholar
  18. 18.
    R.M. Arkhipov, T. Habruseva, A.S. Pimenov, M. Radziunas, S.P. Hegarty, G. Huyet, A.G. Vladimirov, J. Opt. Soc. Am. B 33, 351 (2016) ADSCrossRefGoogle Scholar
  19. 19.
    S. Breuer, W. Elsäßer, J.G. McInerney, K. Yvind, J. Pozo, E.A.J.M. Bente, M. Yousefi, A. Villafranca, N. Vogiatzis, J. Rorison, IEEE J. Quantum Electr. 46, 150 (2010) ADSCrossRefGoogle Scholar
  20. 20.
    C. Otto, K. Lüdge, A.G. Vladimirov, M. Wolfrum, E. Schöll, New J. Phys. 14, 113033 (2012) ADSCrossRefGoogle Scholar
  21. 21.
    L. Drzewietzki, S. Breuer, W. Elsäßer, Opt. Express 21, 16142 (2013) ADSCrossRefGoogle Scholar
  22. 22.
    C. Otto, L.C. Jaurigue, E. Schöll, K. Lüdge, IEEE Photonics J. 6, 1501814 (2014) CrossRefGoogle Scholar
  23. 23.
    L.C. Jaurigue, A.S. Pimenov, D. Rachinskii, E. Schöll, K. Lüdge, A.G. Vladimirov, Phys. Rev. A 92, 053807 (2015) ADSCrossRefGoogle Scholar
  24. 24.
    L.C. Jaurigue, O. Nikiforov, E. Schöll, S. Breuer, K. Lüdge, Phys. Rev. E 93, 022205 (2016) ADSCrossRefGoogle Scholar
  25. 25.
    V.Z. Tronciu, H.J. Wünsche, M. Wolfrum, M. Radziunas, Phys. Rev. E 73, 046205 (2006) ADSCrossRefGoogle Scholar
  26. 26.
    M. Radziunas, A. Glitzky, U. Bandelow, M. Wolfrum, U. Troppenz, J. Kreissl, W. Rehbein, IEEE J. Sel. Top. Quantum Electr. 13, 136 (2007) ADSCrossRefGoogle Scholar
  27. 27.
    A.G. Vladimirov, D.V. Turaev, Phys. Rev. A 72, 033808 (2005) ADSCrossRefGoogle Scholar
  28. 28.
    L.C. Jaurigue, B. Krauskopf, K. Lüdge, Chaos 27, 114301 (2017) ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    L.C. Jaurigue, Passively Mode-Locked Semiconductor Lasers: Dynamics and Stochastic Properties in the Presenceof Optical Feedback (Springer, 2017) Google Scholar
  30. 30.
    A.G. Vladimirov, D.V. Turaev, G. Kozyreff, Opt. Lett. 29, 1221 (2004) ADSCrossRefGoogle Scholar
  31. 31.
    C. Otto, Dynamics of Quantum Dot Lasers – Effects of Optical Feedback and External Optical Injection (Springer, Heidelberg, 2014) Google Scholar
  32. 32.
    J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations (Springer, New York, 1993) Google Scholar
  33. 33.
    R. Szalai, PDDE-CONT:   A   continuation   and   bifurcation  software  for  delay-differential  equations (Budapest University of Technology and Economics, Hungary, 2007) Google Scholar
  34. 34.
    R. Szalai, KNUT: A continuation and bifurcation software for delay-differential equations, 2009, Available at
  35. 35.
    D. Roose, R. Szalai, Continuation and bifurcation analysis of delay differential equations, in Numerical Continuation Methods for Dynamical Systems (Springer, 2007), p. 359 Google Scholar
  36. 36.
    K. Engelborghs, T. Luzyanina, D. Roose, ACM Trans. Math. Softw. 28, 1 (2002) CrossRefGoogle Scholar
  37. 37.
    K. Engelborghs, T. Luzyanina, D. Roose, J. Comput. Appl. Math. 125, 265 (2000) ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    S. Terrien, B. Krauskopf, N.G.R. Broderick, L.C. Jaurigue, K. Lüdge, Phys. Rev. A 98, 043819 (2018) ADSCrossRefGoogle Scholar
  39. 39.
    B. Krauskopf, K. Green, J. Comp. Phys. 186, 230 (2003) ADSCrossRefGoogle Scholar
  40. 40.
    R.C. Calleja, A.R. Humphries, B. Krauskopf, SIAM J. Appl. Dyn. Syst. 16, 1474 (2017) MathSciNetCrossRefGoogle Scholar
  41. 41.
    S. Yanchuk, P. Perlikowski, Phys. Rev. E 79, 046221 (2009) ADSMathSciNetCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Institute of Theoretical Physics, Technische Universität BerlinBerlinGermany
  2. 2.Dodd Walls Centre for Photonic and Quantum Technologies, Department of Mathematics, University of AucklandAucklandNew Zealand

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