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

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2013 (4)

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]

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]

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]

2012 (2)

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. 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]

2011 (3)

J. Fell, J. Axmacher, “The role of phase synchronization in memory processes,” Nat. Rev. Neurosci. 12, 105–118 (2011).
[CrossRef] [PubMed]

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]

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

2010 (2)

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]

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]

2009 (2)

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]

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]

2008 (1)

A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, “Synchronization in complex networks,” Phys. Rep. 469, 93–153 (2008).
[CrossRef]

2007 (2)

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]

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]

2005 (1)

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

2004 (1)

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]

1997 (1)

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]

1996 (1)

K. Wiesenfeld, P. Colet, S. H. Strogatz, “Synchronization transition in a disordered Josephson series array,” Phys. Rev. Lett. 76, 404–407 (1996).
[CrossRef] [PubMed]

1994 (1)

H. G. Solari, G.-L. Oppo, “Laser with injected signal: perturbation of an invariant circle,” Opt. Commun. 111, 173–190 (1994).
[CrossRef]

1990 (2)

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]

D. G. Aronson, G. B. Ermentrout, N. Kopell, “Amplitude response of coupled oscillators,” Physica D 41, 403–449 (1990).
[CrossRef]

1988 (1)

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]

1982 (1)

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]

Agrawal, D. K.

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]

Arenas, A.

A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, “Synchronization in complex networks,” Phys. Rep. 469, 93–153 (2008).
[CrossRef]

Aronson, D. G.

D. G. Aronson, G. B. Ermentrout, N. Kopell, “Amplitude response of coupled oscillators,” Physica D 41, 403–449 (1990).
[CrossRef]

Axmacher, J.

J. Fell, J. Axmacher, “The role of phase synchronization in memory processes,” Nat. Rev. Neurosci. 12, 105–118 (2011).
[CrossRef] [PubMed]

Baselga Pascual, B.

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]

Baselga-Pascual, B.

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]

Blanc, S.

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]

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]

Braza, P. A.

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]

Bretenaker, F.

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]

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]

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]

Brisset, J.

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]

Brunel, M.

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]

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]

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]

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]

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]

Chakraborty, T.

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]

Chow, W. W.

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]

Chu, C.

Colet, P.

K. Wiesenfeld, P. Colet, S. H. Strogatz, “Synchronization transition in a disordered Josephson series array,” Phys. Rev. Lett. 76, 404–407 (1996).
[CrossRef] [PubMed]

Crozatier, V.

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]

Czeisler, C. A.

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]

Diaz-Guilera, A.

A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, “Synchronization in complex networks,” Phys. Rep. 469, 93–153 (2008).
[CrossRef]

Dolfi, D.

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]

Emile, O.

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]

Epstein, I. R.

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]

Ermentrout, G. B.

D. G. Aronson, G. B. Ermentrout, N. Kopell, “Amplitude response of coupled oscillators,” Physica D 41, 403–449 (1990).
[CrossRef]

Erneux, T.

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]

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]

T. Erneux, P. Glorieux, Laser Dynamics (Cambridge University, 2010).
[CrossRef]

T. Erneux, Applied Delay Differential Equations, (Springer, 2009).

Fell, J.

J. Fell, J. Axmacher, “The role of phase synchronization in memory processes,” Nat. Rev. Neurosci. 12, 105–118 (2011).
[CrossRef] [PubMed]

Ferrand, B.

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]

Fischer, I.

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]

Fraden, S.

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]

García-Ojalvo, J.

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]

Gatare, I.

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]

Gilles, H.

Girard, S.

Glorieux, P.

T. Erneux, P. Glorieux, Laser Dynamics (Cambridge University, 2010).
[CrossRef]

Gonzalez-Ochoa, H. O.

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]

Goulding, D.

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]

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]

Grillot, F.

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]

Hegarty, S. P.

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]

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]

Heinrich, G.

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

Huignard, J.-P.

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]

Hung, Y.

Huyet, G.

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]

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]

Hwang, S.

Kelleher, B.

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]

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]

Kervevan, L.

Kopell, N.

D. G. Aronson, G. B. Ermentrout, N. Kopell, “Amplitude response of coupled oscillators,” Physica D 41, 403–449 (1990).
[CrossRef]

Kovanis, V.

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]

Krauskopf, B.

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

Kronauer, R. E.

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]

Kubala, B.

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

Kurths, J.

A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, “Synchronization in complex networks,” Phys. Rep. 469, 93–153 (2008).
[CrossRef]

A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University, 2003).

Lai, N. D.

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]

Laroche, M.

Le Floch, A.

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]

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]

Lenstra, D.

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

Lester, L. F.

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]

Lingnau, B.

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]

Locquet, A.

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]

Lüdge, Kathy

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]

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]

Ludwig, M.

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

Majer, N.

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]

Marquardt, F.

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

Merlet, T.

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]

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]

Mihara, T.

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]

Mirasso, C. R.

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]

Molva, E.

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]

Moore-Ede, M. C.

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]

Moreno, Y.

A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, “Synchronization in complex networks,” Phys. Rep. 469, 93–153 (2008).
[CrossRef]

Morvan, L.

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]

Naderi, N. A.

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]

Oppo, G.-L.

H. G. Solari, G.-L. Oppo, “Laser with injected signal: perturbation of an invariant circle,” Opt. Commun. 111, 173–190 (1994).
[CrossRef]

Otto, C.

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]

Ozaki, M.

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]

Panajotov, K.

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]

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]

Pausch, J.

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]

Pikovsky, A.

A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University, 2003).

Pilato, S. F.

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]

Pochet, M.

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]

Poezevara, A.

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]

Qian, J.

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

Rand, R. H.

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]

Romanelli, M.

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]

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]

Rosenblum, M.

A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University, 2003).

Rubiola, E.

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

Schöll, E.

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]

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]

Sciamanna, M.

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]

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]

Seshia, A. A.

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]

Simpson, T. B.

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

Solari, H. G.

H. G. Solari, G.-L. Oppo, “Laser with injected signal: perturbation of an invariant circle,” Opt. Commun. 111, 173–190 (1994).
[CrossRef]

Someya, H.

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]

Soriano, M. C.

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]

Strogatz, S. H.

K. Wiesenfeld, P. Colet, S. H. Strogatz, “Synchronization transition in a disordered Josephson series array,” Phys. Rev. Lett. 76, 404–407 (1996).
[CrossRef] [PubMed]

S. H. Strogatz, Sync: How Order Emerges from Chaos in the Universe, Nature and Daily Life (Hyperion, 2003).

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]

Terry, N. B.

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]

Thévenin, J.

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]

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]

Toiya, M.

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]

Tylaite, E.

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]

Uchida, A.

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]

Vallet, M.

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]

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]

Vanag, V. K.

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]

Weitzman, E. D.

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]

Wieczorek, S.

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

Wiesenfeld, K.

K. Wiesenfeld, P. Colet, S. H. Strogatz, “Synchronization transition in a disordered Josephson series array,” Phys. Rev. Lett. 76, 404–407 (1996).
[CrossRef] [PubMed]

Woodhouse, J.

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]

Yeung, M. K. S.

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]

Yoshimori, S.

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]

Zhou, C.

A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, “Synchronization in complex networks,” Phys. Rep. 469, 93–153 (2008).
[CrossRef]

Am. J. Physiol. (1)

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]

Eur. Phys. J. D (1)

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]

IEEE J. Sel. Top. Quantum Electron. (1)

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]

IEEE Photon. Technol. Lett. (1)

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]

Int. J. Non-Linear Mech. (1)

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]

J. Phys. Chem. Lett. (1)

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]

Nat. Rev. Neurosci. (1)

J. Fell, J. Axmacher, “The role of phase synchronization in memory processes,” Nat. Rev. Neurosci. 12, 105–118 (2011).
[CrossRef] [PubMed]

New Journal of Physics (2)

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]

Opt. Commun. (1)

H. G. Solari, G.-L. Oppo, “Laser with injected signal: perturbation of an invariant circle,” Opt. Commun. 111, 173–190 (1994).
[CrossRef]

Opt. Lett. (2)

Optical Review (1)

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]

Phys. Rep. (2)

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]

Phys. Rev. A (1)

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]

Phys. Rev. E (3)

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]

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]

Phys. Rev. Lett. (4)

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]

K. Wiesenfeld, P. Colet, S. H. Strogatz, “Synchronization transition in a disordered Josephson series array,” Phys. Rev. Lett. 76, 404–407 (1996).
[CrossRef] [PubMed]

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]

Physica D (1)

D. G. Aronson, G. B. Ermentrout, N. Kopell, “Amplitude response of coupled oscillators,” Physica D 41, 403–449 (1990).
[CrossRef]

Rev. Mod. Phys. (1)

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]

Other (9)

T. Erneux, Applied Delay Differential Equations, (Springer, 2009).

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]

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. 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]

T. Erneux, P. Glorieux, Laser Dynamics (Cambridge University, 2010).
[CrossRef]

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).

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Figures (5)

Fig. 1
Fig. 1

(a) Dual-Frequency Laser. M1,2: cavity mirrors. QWP: quarter-wave plate. (b) Experimental setup that allows synchronizing the DFL beat-note frequency to a RF synthesizer. DFL: Dual-frequency laser (slave oscillator). M: feedback mirror. QWP: quarter-wave plate. AOM: acousto-optic modulator. P: polarizer. D: detector. P-Q analyzer: digital vector signal analyzer, permitting to measure the quadratures of I with respect to the reference signal delivered by the RF synthesizer.

Fig. 2
Fig. 2

(a) Experimental phasor plots of the output signal I = |Ex + Ey|2, in the reference frame of the local oscillator at 2fAO. Each plot contains 2000 points, recorded over 100 μs. (I) Phase-locking regime. (II) Bounded-phase regime. (III) Unbounded-phase regime. (b) Corresponding experimental time series. (c) Power spectra.

Fig. 3
Fig. 3

Measured phase spectra Sϕ (f) for different values of the detuning Δν.

Fig. 4
Fig. 4

(a) Measured phase spectra Sϕ (f) at 1 kHz from the carrier frequency, as a function of the detuning Δν. The arrows correspond to the spectra of Fig. 3 (except for the free-running laser situation, not represented here). The red line is an eye-guide. (b) A phase bifurcation diagram, calculated using the laser model and the measured values of fA and fB, is shown for comparison.

Fig. 5
Fig. 5

(a) Simulated phase spectra Sϕ (f) using the laser model. Black: Δν = 100 kHz, phase locking regime. Blue: Δν = 180 kHz, bounded-phase regime. Violet: Δν = 230 kHz ≃ fB. Red: Δν = 400 kHz, phase-drifting regime. Green: free-running laser (no feedback from the external cavity). The values of the parameters used in the simulations are: β = 0.6, fA = 160 kHz, fR = 70 kHz, γ = fA/fR, ε = 0.97 10−2, η = 1.2. (b) Simulated phase spectra Sϕ (f) for the generic model. Black: e = 1.7, Δ = 1.4, phase locking regime. Blue: e = 1.7, Δ = 2.6, bounded-phase regime. Violet: e = 1.7, Δ = 2.9 ≃ fB. Red: e = 1.7, Δ = 5.7, phase-drifting regime. Green: e = 0, no forcing.

Equations (6)

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C ( τ ) = ϕ ( t ) ϕ ( t + τ ) = lim T 1 T T / 2 T / 2 ϕ ( t ) ϕ ( t + τ ) d t .
PSD ( f ) = 2 θ ( f ) + C ( τ ) exp ( i 2 π f τ ) d τ ,
d e x d s = ( m x + β m y ) 1 + β e x 2 ,
d e y d s = ( m y + β m x ) 1 + β e y 2 + i Δ e y + γ e x ,
d m x , y d s = 1 ( | e x , y | 2 + β | e y , x | 2 ) + ε m x , y [ 1 + ( η 1 ) ( | e x , y | 2 + β | e y , x | 2 ) ] ,
d y d s = ( 1 i Δ ) y ( 1 + i α ) | y | 2 y + e .

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