OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 7 — Apr. 7, 2014
  • pp: 7364–7373

Measuring the universal synchronization properties of driven oscillators across a Hopf instability

M. Romanelli, L. Wang, M. Brunel, and M. Vallet  »View Author Affiliations

Optics Express, Vol. 22, Issue 7, pp. 7364-7373 (2014)

View Full Text Article

Enhanced HTML    Acrobat PDF (4213 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



When a driven oscillator loses phase-locking to a master oscillator via a Hopf bifurcation, it enters a bounded-phase regime in which its average frequency is still equal to the master frequency, but its phase displays temporal oscillations. Here we characterize these two synchronization regimes in a laser experiment, by measuring the spectrum of the phase fluctuations across the bifurcation. We find experimentally, and confirm numerically, that the low frequency phase noise of the driven oscillator is strongly suppressed in both regimes in the same way. Thus the long-term phase stability of the master oscillator is transferred to the driven one, even in the absence of phase-locking. The numerical study of a generic, minimal model suggests that such behavior is universal for any periodically driven oscillator near a Hopf bifurcation point.

© 2014 Optical Society of America

OCIS Codes
(140.3520) Lasers and laser optics : Lasers, injection-locked
(140.3580) Lasers and laser optics : Lasers, solid-state
(190.3100) Nonlinear optics : Instabilities and chaos

ToC Category:
Nonlinear Optics

Original Manuscript: January 21, 2014
Revised Manuscript: March 5, 2014
Manuscript Accepted: March 6, 2014
Published: March 24, 2014

Virtual Issues
Physics and Applications of Laser Dynamics (2014) Optics Express

M. Romanelli, L. Wang, M. Brunel, and M. Vallet, "Measuring the universal synchronization properties of driven oscillators across a Hopf instability," Opt. Express 22, 7364-7373 (2014)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. S. H. Strogatz, Sync: How Order Emerges from Chaos in the Universe, Nature and Daily Life (Hyperion, 2003).
  2. A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University, 2003).
  3. K. Wiesenfeld, P. Colet, S. H. Strogatz, “Synchronization transition in a disordered Josephson series array,” Phys. Rev. Lett. 76, 404–407 (1996). [CrossRef] [PubMed]
  4. M. Toiya, H. O. Gonzalez-Ochoa, V. K. Vanag, S. Fraden, I. R. Epstein, “Synchronization of chemical micro-oscillators,” J. Phys. Chem. Lett. 1, 1241–1246 (2010). [CrossRef]
  5. J. Fell, J. Axmacher, “The role of phase synchronization in memory processes,” Nat. Rev. Neurosci. 12, 105–118 (2011). [CrossRef] [PubMed]
  6. A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, “Synchronization in complex networks,” Phys. Rep. 469, 93–153 (2008). [CrossRef]
  7. S. Wieczorek, B. Krauskopf, T. B. Simpson, D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. 416, 1–128 (2005). [CrossRef]
  8. T. Erneux, P. Glorieux, Laser Dynamics (Cambridge University, 2010). [CrossRef]
  9. N. A. Naderi, M. Pochet, F. Grillot, N. B. Terry, V. Kovanis, L. F. Lester, “Modeling the injection-locked behavior of a quantum dash semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 15, 563–571 (2009). [CrossRef]
  10. Y. Hung, C. Chu, S. Hwang, “Optical double-sideband modulation to single-sideband modulation conversion using period-one nonlinear dynamics of semiconductor lasers for radio-over-fiber links,” Opt. Lett. 38, 1482–1484 (2013). [CrossRef] [PubMed]
  11. B. Kelleher, D. Goulding, B. Baselga-Pascual, S. P. Hegarty, G. Huyet, “Phasor plots in optical injection experiments,” Eur. Phys. J. D 58, 175–179 (2010). [CrossRef]
  12. J. Thévenin, M. Romanelli, M. Vallet, M. Brunel, T. Erneux, “Resonance assisted synchronization of coupled oscillators: frequency locking without phase locking,” Phys. Rev. Lett. 107, 104101 (2011). [CrossRef] [PubMed]
  13. P. A. Braza, T. Erneux, “Constant phase, phase drift, and phase entrainment in lasers with an injected signal,” Phys. Rev. A 41, 6470–6479 (1990). [CrossRef] [PubMed]
  14. H. G. Solari, G.-L. Oppo, “Laser with injected signal: perturbation of an invariant circle,” Opt. Commun. 111, 173–190 (1994). [CrossRef]
  15. M. K. S. Yeung, S. H. Strogatz, “Nonlinear dynamics of a solid-state laser with injection,” Phys. Rev. E58, 4421–4435 (1998); M. K. S. Yeung and S. H. Strogatz, “Erratum: Nonlinear dynamics of a solid-state laser with injection,” Phys. Rev. E 61, 2154–2154 (2000). [CrossRef]
  16. J. Pausch, C. Otto, E. Tylaite, N. Majer, E. Schöll, Kathy Lüdge, “Optically injected quantum dot lasers: impact of nonlinear carrier lifetimes on frequency-locking dynamics,” New Journal of Physics 14, 053018 (2012). [CrossRef]
  17. B. Lingnau, W. W. Chow, E. Schöll, Kathy Lüdge, “Feedback and injection locking instabilities in quantum-dot lasers: a microscopically based bifurcation analysis,” New Journal of Physics 15, 093031 (2013). [CrossRef]
  18. R. E. Kronauer, C. A. Czeisler, S. F. Pilato, M. C. Moore-Ede, E. D. Weitzman, “Mathematical model of the human circadian system with two interacting oscillators,” Am. J. Physiol. 242, R3–R17 (1982). [PubMed]
  19. T. Chakraborty, R. H. Rand, “The transition from phase locking to drift in a system of two weakly coupled van der Pol oscillators,” Int. J. Non-Linear Mech. 23, 369–376 (1988). [CrossRef]
  20. B. Kelleher, D. Goulding, B. Baselga Pascual, S. P. Hegarty, G. Huyet, “Bounded phase phenomena in the optically injected laser,” Phys. Rev. E 85, 046212 (2012). [CrossRef]
  21. D. G. Aronson, G. B. Ermentrout, N. Kopell, “Amplitude response of coupled oscillators,” Physica D 41, 403–449 (1990). [CrossRef]
  22. J. Thévenin, M. Romanelli, M. Vallet, M. Brunel, T. Erneux, “Phase and intensity dynamics of a two-frequency laser submitted to resonant frequency-shifted feedback,” Phys. Rev. A86, 033815 (2012). [CrossRef]
  23. M. Brunel, N. D. Lai, M. Vallet, A. Le Floch, F. Bretenaker, L. Morvan, D. Dolfi, J.-P. Huignard, S. Blanc, T. Merlet, “Generation of tunable high-purity microwave and terahertz signals by two-frequency solid state lasers,” Proc. SPIE 5466, Microwave and Terahertz Photonics, 131–139 (2004). [CrossRef]
  24. M. Brunel, O. Emile, F. Bretenaker, A. Le Floch, B. Ferrand, E. Molva, “Tunable two-frequency lasers for lifetime measurements,” Optical Review 4, 550–552 (1997). [CrossRef]
  25. L. Kervevan, H. Gilles, S. Girard, M. Laroche, “Beat-note jitter suppression in a dual-frequency laser using optical feedback,” Opt. Lett. 32, 1099–1101 (2007). [CrossRef] [PubMed]
  26. E. Rubiola, Phase Noise and Frequency Stability in Oscillators (Cambridge University, 2008). [CrossRef]
  27. IEEE Standard Definitions of Physical Quantities for Fundamental Frequency and Time Metrology, IEEE Standard 1139–2008.
  28. M. Brunel, F. Bretenaker, S. Blanc, V. Crozatier, J. Brisset, T. Merlet, A. Poezevara, “High-spectral purity RF beat note generated by a two-frequency solid-state laser in a dual thermooptic and electrooptic phase-locked loop,” IEEE Photon. Technol. Lett. 16, 870–872 (2004). [CrossRef]
  29. G. Heinrich, M. Ludwig, J. Qian, B. Kubala, F. Marquardt, “Collective Dynamics in Optomechanical Arrays,” Phys. Rev. Lett. 107, 043603 (2011). [CrossRef] [PubMed]
  30. D. K. Agrawal, J. Woodhouse, A. A. Seshia, “Observation of Locked Phase Dynamics and Enhanced Frequency Stability in Synchronized Micromechanical Oscillators,” Phys. Rev. Lett. 111, 084101 (2013). [CrossRef] [PubMed]
  31. M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85, 421–470 (2013). [CrossRef]
  32. T. Erneux, Applied Delay Differential Equations, (Springer, 2009).
  33. M. Sciamanna, I. Gatare, A. Locquet, K. Panajotov, “Polarization synchronization in unidirectionally coupled vertical-cavity surface-emitting lasers with orthogonal optical injection,” Phys. Rev. E 75, 056213 (2007). [CrossRef]
  34. M. Ozaki, H. Someya, T. Mihara, A. Uchida, S. Yoshimori, K. Panajotov, M. Sciamanna, “Leader-laggard relationship of chaos synchronization in mutually coupled vertical-cavity surface-emitting lasers with time delay,” Phys. Rev. E 79, 026210 (2009). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


Fig. 1 Fig. 2 Fig. 3
Fig. 4 Fig. 5

Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited