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

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

http://dx.doi.org/10.1364/OE.22.007364

Acrobat PDF (4213 KB)

### Abstract

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

## 1. Introduction

3. K. Wiesenfeld, P. Colet, and 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, and I. R. Epstein, “Synchronization of chemical micro-oscillators,” J. Phys. Chem. Lett. **1**, 1241–1246 (2010). [CrossRef]

5. J. Fell and 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, and C. Zhou, “Synchronization in complex networks,” Phys. Rep. **469**, 93–153 (2008). [CrossRef]

## 2. Experimental setup and results

### 2.1. Experimental setup

22. J. Thévenin, M. Romanelli, M. Vallet, M. Brunel, and 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, and 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]

*λ*= 1064 nm). The active medium is pumped by a laser diode emitting at 808 nm. A 1 mm-thick silica talon ensures single longitudinal mode oscillation. Two eigenmodes

*E*and

_{x}*E*, polarized along

_{y}*x̂*and

*ŷ*, with eigenfrequencies

*ν*and

_{x}*ν*respectively, oscillate simultaneously. An intracavity birefringent element (here two quarter-wave plates QWPs) induces a frequency difference, finely tunable from 0 to

_{y}*c*/4

*L*= 1 GHz by rotating one QWP with respect to the other [24

24. M. Brunel, O. Emile, F. Bretenaker, A. Le Floch, B. Ferrand, and E. Molva, “Tunable two-frequency lasers for lifetime measurements,” Optical Review **4**, 550–552 (1997). [CrossRef]

*ν*

_{0}=

*ν*−

_{y}*ν*is obtained. The DFL can thus be seen as an opto-RF oscillator.

_{x}25. L. Kervevan, H. Gilles, S. Girard, and M. Laroche, “Beat-note jitter suppression in a dual-frequency laser using optical feedback,” Opt. Lett. **32**, 1099–1101 (2007). [CrossRef] [PubMed]

*x̂*and

*ŷ*polarizations, and finally the laser beam is reinjected in the laser cavity after crossing again the AOM. As a result, a

*x̂*-polarized field oscillating at the frequency

*ν*+ 2

_{y}*f*and a

_{AO}*ŷ*-polarized field oscillating at the frequency

*ν*+ 2

_{x}*f*are reinjected in the laser. We choose the value of 2

_{AO}*f*so that

_{AO}*ν*+ 2

_{x}*f*is close to

_{AO}*ν*. Under suitable feedback conditions,

_{y}*ν*locks to the injected beam frequency

_{y}*ν*+ 2

_{x}*f*, i.e. the frequency difference

_{AO}*ν*−

_{y}*ν*locks to 2

_{x}*f*. We note that the optical reinjection has no direct effect on

_{AO}*E*, because the frequency difference between

_{x}*ν*and

_{x}*ν*+ 2

_{y}*f*is too large. For the same reason, multiple round trips in the feedback cavity have no effect on the dynamics. The laser output is detected with a photodiode (3 GHz analogic bandwidth) after a crossed polarizer, thus providing an electrical signal proportional to

_{AO}*I*= |

*E*+

_{x}*E*|

_{y}^{2}. The signal is then analyzed with an electrical spectrum analyzer, a digital P-Q signal analyzer and an oscilloscope.

### 2.2. Experimental results

*ν*

_{0}=

*ν*−

_{y}*ν*around 200 MHz, is obtained by detecting the interference signal

_{x}*I*= |

*E*+

_{x}*E*|

_{y}^{2}between the laser fields on a photodiode after a polarizer. The whole system can thus be seen as an opto-RF oscillator, driven by the RF synthesizer (master oscillator) when the optical feedback is present. The beat-note signal

*I*can be visualized at each instant in time as a vector having two components

*P*=

*I*

_{0}(

*t*)cos

*ϕ*(

*t*) and

*Q*=

*I*

_{0}(

*t*)sin

*ϕ*(

*t*) in a reference frame rotating at the master signal frequency 2

*f*. The phase

_{AO}*ϕ*(

*t*) is defined as

*ϕ*(

*t*) =

*ϕ*(

_{y}*t*) −

*ϕ*(

_{x}*t*) − 2

*π*2

*f*, where

_{AO}t*ϕ*are the optical phases of the two electric fields

_{x,y}*E*and

_{x}*E*. For small enough detuning Δ

_{y}*ν*= Δ

*ν*

_{0}− 2

*f*between the slave and the master oscillator, phase-locking occurs, and the corresponding experimental phasor plot consists of just a fixed point (Fig. 2(a), regime I). If Δ

_{AO}*ν*is increased above the Adler frequency

*f*, the fixed point solution loses stability through a Hopf bifurcation. The point representing

_{A}*I*then rotates at the frequency Δ

*ν*on a limit cycle (Fig. 2(a), regime II). The relative phase exhibits bounded, periodic temporal oscillations, whose amplitude increases with Δ

*ν*. Finally, when Δ

*ν*is bigger than a second characteristic frequency

*f*, the origin of the plane lies inside the limit cycle and the phase variations are not bounded anymore (Fig. 2(a), regime III). In the time and frequency domains (Fig. 2(b–c)), phase-locking (I) appears as pure sinusoidal oscillation, and the corresponding power spectrum consists of a single, sharp Fourier peak. On the contrary, in the bounded (II) and in the unbounded-phase regimes (III), the power spectrum shows two peaks at the natural frequencies 2

_{B}*f*and Δ

_{AO}*ν*

_{0}of the uncoupled master and slave oscillator respectively, and the time series displays a beating, i.e. a slow modulation of the amplitude of the oscillations.

*f*< Δ

_{A}*ν*<

*f*. On the one hand, since the relative phase is bounded, the average frequency of the oscillators is the same, and they can be considered as synchronized. On the other hand, the bounded-phase regime appears very similar to the unbounded-phase regime. In particular, the power spectrum displays two separated peaks at the natural frequencies of the oscillators, as it happens when synchronization is absent. Furthermore, while a qualitative change of the dynamics occurs when passing from phase locking to the bounded phase through a Hopf bifurcation, in the {P, Q} plane in Fig. 2(a) the bounded-phase and unbounded-phase regimes appear topologically equivalent and no bifurcation is seen between them [7

_{B}7. S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. **416**, 1–128 (2005). [CrossRef]

26. E. Rubiola, *Phase Noise and Frequency Stability in Oscillators* (Cambridge University, 2008). [CrossRef]

*ϕ*(

*t*), the autocorrelation function is: Brackets indicate the ensemble average. The single-sideband power spectral density of the process is defined as where

*θ*(

*f*) is the Heaviside function. Indeed, the negative frequency components can be discarded because

*C*(

*τ*) is a real, even function.

*PSD*(

*f*) is frequently expressed in decibels as 10 log[

*PSD*(

*f*)], and its units are dBrad

^{2}/Hz. This is the quantity we have measured, and indicated as

*S*(

_{ϕ}*f*) = 10log[

*PSD*(

*f*)]. In practice, the above definitions are not directly used for the computation of

*S*(

_{ϕ}*f*). One rather uses the equality

*PSD*(

_{ϕ}*f*)

*df*= 2

*θ*(

*f*) 〈|

*d*Φ(

*f*)|

^{2}〉, and estimates |

*d*Φ(

*f*)|

^{2}by taking the square modulus of the Fast-Fourier Transform (FFT) of a sampled time series of

*ϕ*(

*t*). In our setup, the relative phase

*ϕ*(

*t*) can be extracted from the time series of the two quadratures

*P*(

*t*) and

*Q*(

*t*). The spectrum

*S*(

_{ϕ}*f*) of

*ϕ*(

*t*) is then computed. Note that, when the phase is unbounded, the average slope of the drifting phase has to be substracted in order to get a stationary, ergodic process. The experimental spectra are shown in Fig. 3. One sees that, in the phase-locking regime, the phase fluctuations of the free running laser are strongly suppressed at low frequency. For instance, at 1 kHz from the carrier one has a noise reduction of more than 30 dB with respect to the free-running case. The phase of the slave oscillator is locked to the stable master, which ensures the long-term phase stability. The absolute value of the phase noise that we obtain, −100 dBrad

^{2}/Hz at 10 kHz from the carrier, is comparable to what is obtained with active stabilization techniques using electronic servo-loops [28

28. M. Brunel, F. Bretenaker, S. Blanc, V. Crozatier, J. Brisset, T. Merlet, and 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]

*ν*, and its harmonics. These peaks are the coherent signature of the Hopf bifurcation, and correspond to the onset of the deterministic oscillation of the phase beyond the bifurcation point

*f*. At the transition between the bounded and unbounded-phase regime, for Δ

_{A}*ν*≃

*f*, there is an enormous increase of the phase noise. This can be understood by observing in Fig. 2(a) that the experimental limit cycle has a finite thickness, because of uncontrolled fluctuations of the parameters and of the setup. Therefore, when Δ

_{B}*ν*approaches

*f*, erratic phase slips occur in a random fashion, driven by the noise present in the system. These stochastic, abrupt 2

_{B}*π*phase jumps close to the point

*f*explain the huge amount of phase noise. Finally, the spectrum of the unbounded-phase regime corresponds essentially to the one of the free-running opto-RF oscillator, apart from the high-frequency peaks. Figure 4(a) reports the PSD value at 1 kHz from the carrier as a function of Δ

_{B}*ν*. It is striking to compare this plot to a numerically calculated bifurcation diagram (Fig. 4(b)). Indeed, when looking at the long-term phase stability of the oscillator, the occurrence of a Hopf bifurcation at the point

*f*is completely undetectable. Conversely, at the point

_{A}*f*a sharp transition occurs as a signature of the unlocking of the slave oscillator. The conclusion that can be drawn is that the relevant boundary for the synchronization range is not the Adler frequency

_{B}*f*, but the bounded-phase frequency

_{A}*f*. From the viewpoint of dynamical systems, phase-locking and bounded-phase are two qualitatively different regimes, separated by a bifurcation; yet this distinction turns out to be of little relevance if one is concerned with synchronization.

_{B}7. S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. **416**, 1–128 (2005). [CrossRef]

*I*

_{0}(

*t*) and the phase

*ϕ*(

*t*), then its phase space is the surface of a half-cylinder. Bounded-phase and unbounded-phase regimes correspond to limit cycles that do not wrap the half-cylinder (librations) or that wrap it (rotations) respectively, and are not topologically equivalent. On the contrary, if the signal is described using the quadratures

*P*(

*t*) and

*Q*(

*t*), the two regimes are topologically equivalent, as we have seen. This difference stems from the fact that the polar coordinates

*I*

_{0}(

*t*) and

*ϕ*(

*t*) are not homeomorphic with the planar coordinates

*P*(

*t*) and

*Q*(

*t*). So, mathematically the nature of the bounded/unbounded-phase transition is, to some extent, a matter of choice, and it is worth noticing that, in a theoretical bifurcation analysis, it looks more convenient to work with

*P*(

*t*) and

*Q*(

*t*), so that no bifurcation appears at

*f*[7

_{B}7. S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. **416**, 1–128 (2005). [CrossRef]

*f*, namely the transition from synchronization to desynchronization.

_{B}## 3. Theory: laser and generic oscillator models

*y*represents the complex amplitude of a periodically driven oscillator, Δ is the detuning between the master and the slave,

*α*accounts for the possible dependence of the oscillator’s phase on the amplitude of the oscillations, and

*e*is the strength of the drive. Equation 6 contains just a linear term accounting for the growth of oscillations, a saturating nonlinearity, and an external forcing, and is a universal equation describing any periodically driven oscillator close to a supercritical Hopf bifurcation point [2]. We have introduced some noise by replacing Δ with Δ[1 + 0.033

*ξ*(

*s*)]. In the following, we have taken

*α*= 0 because this seems more appropriate for comparison with a solid-state laser. However, some numerical simulations, not shown here, indicate that this parameter does not influence phase noise, when

*α*

^{2}< 1/3. For higher values of

*α*, the bifurcation structure of the generic model changes qualitatively [2], and we have not explored this case. The PSDs of the phase of

*y*are plotted in Fig. 5(b), for different values of Δ. The agreement of the spectra in Fig. 5(b) with the laser model is striking. All the observed regimes are reproduced and appear thus to be generic for any driven oscillator undergoing a Hopf bifurcation.

## 4. Conclusion

15. M. K. S. Yeung and 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]

29. G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, “Collective Dynamics in Optomechanical Arrays,” Phys. Rev. Lett. **107**, 043603 (2011). [CrossRef] [PubMed]

30. D. K. Agrawal, J. Woodhouse, and 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, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. **85**, 421–470 (2013). [CrossRef]

*f*≃ 10

_{R}*μ*s, which is much longer than the external-cavity round-trip time

*τ*≃ 5 ns. On the contrary, the effect of delay could be conveniently studied using semiconductor lasers, which evolve on much faster time scales [33

33. M. Sciamanna, I. Gatare, A. Locquet, and 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, and 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]

## Acknowledgments

## References and links

1. | S. H. Strogatz, |

2. | A. Pikovsky, M. Rosenblum, and J. Kurths, |

3. | K. Wiesenfeld, P. Colet, and S. H. Strogatz, “Synchronization transition in a disordered Josephson series array,” Phys. Rev. Lett. |

4. | M. Toiya, H. O. Gonzalez-Ochoa, V. K. Vanag, S. Fraden, and I. R. Epstein, “Synchronization of chemical micro-oscillators,” J. Phys. Chem. Lett. |

5. | J. Fell and J. Axmacher, “The role of phase synchronization in memory processes,” Nat. Rev. Neurosci. |

6. | A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, “Synchronization in complex networks,” Phys. Rep. |

7. | S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. |

8. | T. Erneux and P. Glorieux, |

9. | N. A. Naderi, M. Pochet, F. Grillot, N. B. Terry, V. Kovanis, and L. F. Lester, “Modeling the injection-locked behavior of a quantum dash semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. |

10. | Y. Hung, C. Chu, and 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. |

11. | B. Kelleher, D. Goulding, B. Baselga-Pascual, S. P. Hegarty, and G. Huyet, “Phasor plots in optical injection experiments,” Eur. Phys. J. D |

12. | J. Thévenin, M. Romanelli, M. Vallet, M. Brunel, and T. Erneux, “Resonance assisted synchronization of coupled oscillators: frequency locking without phase locking,” Phys. Rev. Lett. |

13. | P. A. Braza and T. Erneux, “Constant phase, phase drift, and phase entrainment in lasers with an injected signal,” Phys. Rev. A |

14. | H. G. Solari and G.-L. Oppo, “Laser with injected signal: perturbation of an invariant circle,” Opt. Commun. |

15. | M. K. S. Yeung and 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 |

16. | J. Pausch, C. Otto, E. Tylaite, N. Majer, E. Schöll, and Kathy Lüdge, “Optically injected quantum dot lasers: impact of nonlinear carrier lifetimes on frequency-locking dynamics,” New Journal of Physics |

17. | B. Lingnau, W. W. Chow, E. Schöll, and Kathy Lüdge, “Feedback and injection locking instabilities in quantum-dot lasers: a microscopically based bifurcation analysis,” New Journal of Physics |

18. | R. E. Kronauer, C. A. Czeisler, S. F. Pilato, M. C. Moore-Ede, and E. D. Weitzman, “Mathematical model of the human circadian system with two interacting oscillators,” Am. J. Physiol. |

19. | T. Chakraborty and 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. |

20. | B. Kelleher, D. Goulding, B. Baselga Pascual, S. P. Hegarty, and G. Huyet, “Bounded phase phenomena in the optically injected laser,” Phys. Rev. E |

21. | D. G. Aronson, G. B. Ermentrout, and N. Kopell, “Amplitude response of coupled oscillators,” Physica D |

22. | J. Thévenin, M. Romanelli, M. Vallet, M. Brunel, and 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, and 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, and E. Molva, “Tunable two-frequency lasers for lifetime measurements,” Optical Review |

25. | L. Kervevan, H. Gilles, S. Girard, and M. Laroche, “Beat-note jitter suppression in a dual-frequency laser using optical feedback,” Opt. Lett. |

26. | E. Rubiola, |

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, and 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. |

29. | G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, “Collective Dynamics in Optomechanical Arrays,” Phys. Rev. Lett. |

30. | D. K. Agrawal, J. Woodhouse, and A. A. Seshia, “Observation of Locked Phase Dynamics and Enhanced Frequency Stability in Synchronized Micromechanical Oscillators,” Phys. Rev. Lett. |

31. | M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. |

32. | T. Erneux, |

33. | M. Sciamanna, I. Gatare, A. Locquet, and K. Panajotov, “Polarization synchronization in unidirectionally coupled vertical-cavity surface-emitting lasers with orthogonal optical injection,” Phys. Rev. E |

34. | M. Ozaki, H. Someya, T. Mihara, A. Uchida, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Leader-laggard relationship of chaos synchronization in mutually coupled vertical-cavity surface-emitting lasers with time delay,” Phys. Rev. E |

**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

**History**

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*

**Citation**

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)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-7364

Sort: Year | Journal | Reset

### References

- S. H. Strogatz, Sync: How Order Emerges from Chaos in the Universe, Nature and Daily Life (Hyperion, 2003).
- A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University, 2003).
- K. Wiesenfeld, P. Colet, S. H. Strogatz, “Synchronization transition in a disordered Josephson series array,” Phys. Rev. Lett. 76, 404–407 (1996). [CrossRef] [PubMed]
- 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]
- J. Fell, J. Axmacher, “The role of phase synchronization in memory processes,” Nat. Rev. Neurosci. 12, 105–118 (2011). [CrossRef] [PubMed]
- A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, “Synchronization in complex networks,” Phys. Rep. 469, 93–153 (2008). [CrossRef]
- S. Wieczorek, B. Krauskopf, T. B. Simpson, D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. 416, 1–128 (2005). [CrossRef]
- T. Erneux, P. Glorieux, Laser Dynamics (Cambridge University, 2010). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- H. G. Solari, G.-L. Oppo, “Laser with injected signal: perturbation of an invariant circle,” Opt. Commun. 111, 173–190 (1994). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- D. G. Aronson, G. B. Ermentrout, N. Kopell, “Amplitude response of coupled oscillators,” Physica D 41, 403–449 (1990). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- E. Rubiola, Phase Noise and Frequency Stability in Oscillators (Cambridge University, 2008). [CrossRef]
- IEEE Standard Definitions of Physical Quantities for Fundamental Frequency and Time Metrology, IEEE Standard 1139–2008.
- 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]
- G. Heinrich, M. Ludwig, J. Qian, B. Kubala, F. Marquardt, “Collective Dynamics in Optomechanical Arrays,” Phys. Rev. Lett. 107, 043603 (2011). [CrossRef] [PubMed]
- 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]
- 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]
- T. Erneux, Applied Delay Differential Equations, (Springer, 2009).
- 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]
- 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.

OSA is a member of CrossRef.