Abstract

Optoelectronic oscillators are the subject of extensive research because of the wide variety of associated applications, which include chaos cryptography, ultrastable microwave generation, and neuromorphic computing. The wideband optoelectronic oscillator presents a particular feature allowing for two dynamical time scales to be superimposed, namely, a slow one and a fast one. In this paper, we fully characterize the onset of the slow-scale oscillation in the wideband optoelectronic oscillator. We investigate the dynamics associated to the first Hopf bifurcation and calculate analytically both the amplitude and period of the induced limit-cycle. In particular, we show how the dynamics of the zero-delay case can be used to provide insight into the infinite-dimensional dynamics of the delayed system. Our theoretical results are in very good agreement with the experimental measurements.

© 2014 Optical Society of America

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    [Crossref]
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    [Crossref]
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    [Crossref]
  10. Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, “Chaotic breathers in delayed electro-optical systems,” Phys. Rev. Lett. 95, 203903 (2005).
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  11. Y. K. Chembo, L. Larger, R. Bendoula, and P. Colet, “Effects of gain and bandwidth on the multimode behavior of optoelectronic microwave oscillators,” Opt. Express 16, 9067–9072 (2008).
    [Crossref]
  12. A. B. Cohen, B. Ravoori, T. E. Murphy, and R. Roy, “Using synchronization for prediction of high-dimensional chaotic dynamics,” Phys. Rev. Lett. 101, 154102 (2008).
    [Crossref]
  13. K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Scholl, “Broadband chaos generated by an optoelectronic oscillator,” Phys. Rev. Lett. 104, 113901 (2010).
    [Crossref]
  14. B. Ravoori, A. B. Cohen, J. Sun, A. E. Motter, T. E. Murphy, and R. Roy, “Robustness of optimal synchronization in real networks,” Phys. Rev. Lett. 107, 034102 (2011).
    [Crossref]
  15. L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Strongly asymmetric square waves in a time-delayed system,” Phys. Rev. E 86, 055201(R) (2012).
    [Crossref]
  16. R. Martinenghi, S. Rybalko, M. Jacquot, Y. K. Chembo, and L. Larger, “Photonic nonlinear transient computing with multiple-delay wavelength dynamics,” Phys. Rev. Lett. 108, 244101 (2012).
    [Crossref]
  17. L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. K. Chembo, and L. Larger, “Slow–fast dynamics of a time-delayed electro-optic oscillator,” Phil. Trans. R. Soc. A 371, 20120459 (2013).
    [Crossref]
  18. K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. 30, 257–261 (1979).
    [Crossref]
  19. K. Ikeda, H. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
    [Crossref]
  20. K. Ikeda and M. Matsumoto, “Study of a high-dimensional chaotic attractor,” J. Stat. Phys. 44, 955–983 (1986).
    [Crossref]
  21. Z. H. Wang, “An iteration method for calculating the periodic solution of time-delay systems after a Hopf bifurcation,” Nonlinear Dyn. 53, 1–11 (2008).
    [Crossref]
  22. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields (Springer-Verlag, 1983).
  23. Z. H. Wang and H. Y. Hu, “Pseudo-oscillator analysis of scalar nonlinear time-delay systems near a Hopf bifurcation,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 17, 2805–2814 (2007).
    [Crossref]
  24. Y. K. Chembo, A. Hmima, P.-A. Lacourt, L. Larger, and J. M. Dudley, “Generation of ultralow jitter optical pulses using optoelectronic oscillators with time-lens soliton-assisted compression,” J. Lightwave Technol. 27, 5160–5167 (2009).
    [Crossref]
  25. R. M. Nguimdo, Y. K. Chembo, P. Colet, and L. Larger, “On the phase noise performance of nonlinear double-loop optoelectronic microwave oscillators,” IEEE J. Quantum Electron. 48, 1415–1423 (2012).
    [Crossref]
  26. A. Coillet, R. Henriet, P. Salzenstein, K. P. Huy, L. Larger, and Y. K. Chembo, “Time-domain dynamics and stability analysis of optoelectronic oscillators based on whispering-gallery mode resonators,” IEEE J. Sel. Top. Quantum Electron. 19, 6000112(2013).
    [Crossref]

2013 (4)

L. Larger, “Complexity in electro-optic delay dynamics: modelling, design and applications,” Phil. Trans. R. Soc. A 371, 20120464 (2013).
[Crossref]

M. C. Soriano, J. Garcia-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]

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. K. Chembo, and L. Larger, “Slow–fast dynamics of a time-delayed electro-optic oscillator,” Phil. Trans. R. Soc. A 371, 20120459 (2013).
[Crossref]

A. Coillet, R. Henriet, P. Salzenstein, K. P. Huy, L. Larger, and Y. K. Chembo, “Time-domain dynamics and stability analysis of optoelectronic oscillators based on whispering-gallery mode resonators,” IEEE J. Sel. Top. Quantum Electron. 19, 6000112(2013).
[Crossref]

2012 (3)

R. M. Nguimdo, Y. K. Chembo, P. Colet, and L. Larger, “On the phase noise performance of nonlinear double-loop optoelectronic microwave oscillators,” IEEE J. Quantum Electron. 48, 1415–1423 (2012).
[Crossref]

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Strongly asymmetric square waves in a time-delayed system,” Phys. Rev. E 86, 055201(R) (2012).
[Crossref]

R. Martinenghi, S. Rybalko, M. Jacquot, Y. K. Chembo, and L. Larger, “Photonic nonlinear transient computing with multiple-delay wavelength dynamics,” Phys. Rev. Lett. 108, 244101 (2012).
[Crossref]

2011 (1)

B. Ravoori, A. B. Cohen, J. Sun, A. E. Motter, T. E. Murphy, and R. Roy, “Robustness of optimal synchronization in real networks,” Phys. Rev. Lett. 107, 034102 (2011).
[Crossref]

2010 (1)

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Scholl, “Broadband chaos generated by an optoelectronic oscillator,” Phys. Rev. Lett. 104, 113901 (2010).
[Crossref]

2009 (1)

2008 (4)

Y. K. Chembo, L. Larger, R. Bendoula, and P. Colet, “Effects of gain and bandwidth on the multimode behavior of optoelectronic microwave oscillators,” Opt. Express 16, 9067–9072 (2008).
[Crossref]

A. B. Cohen, B. Ravoori, T. E. Murphy, and R. Roy, “Using synchronization for prediction of high-dimensional chaotic dynamics,” Phys. Rev. Lett. 101, 154102 (2008).
[Crossref]

Y. K. Chembo, L. Larger, and P. Colet, “Nonlinear dynamics and spectral stability of optoelectronic oscillators,” IEEE J. Quantum Electron. 44, 858–866 (2008).
[Crossref]

Z. H. Wang, “An iteration method for calculating the periodic solution of time-delay systems after a Hopf bifurcation,” Nonlinear Dyn. 53, 1–11 (2008).
[Crossref]

2007 (2)

Z. H. Wang and H. Y. Hu, “Pseudo-oscillator analysis of scalar nonlinear time-delay systems near a Hopf bifurcation,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 17, 2805–2814 (2007).
[Crossref]

Y. K. Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola, and P. Colet, “Dynamic instabilities of microwaves generated with optoelectronic oscillators,” Opt. Lett. 32, 2571–2573 (2007).
[Crossref]

2005 (2)

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438, 343–346 (2005).
[Crossref]

Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, “Chaotic breathers in delayed electro-optical systems,” Phys. Rev. Lett. 95, 203903 (2005).
[Crossref]

2002 (1)

J.-P. Goedgebuer, P. Levy, L. Larger, C.-C. Chen, and W. T. Rhodes, “Optical communication with synchronized hyperchaos generated electrooptically,” IEEE J. Quantum Electron. 38, 1178–1183 (2002).
[Crossref]

1996 (1)

1986 (1)

K. Ikeda and M. Matsumoto, “Study of a high-dimensional chaotic attractor,” J. Stat. Phys. 44, 955–983 (1986).
[Crossref]

1980 (1)

K. Ikeda, H. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
[Crossref]

1979 (1)

K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. 30, 257–261 (1979).
[Crossref]

Akimoto, O.

K. Ikeda, H. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
[Crossref]

Annovazzi-Lodi, V.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438, 343–346 (2005).
[Crossref]

Argyris, A.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438, 343–346 (2005).
[Crossref]

Bendoula, R.

Callan, K. E.

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Scholl, “Broadband chaos generated by an optoelectronic oscillator,” Phys. Rev. Lett. 104, 113901 (2010).
[Crossref]

Chembo, Y.

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Strongly asymmetric square waves in a time-delayed system,” Phys. Rev. E 86, 055201(R) (2012).
[Crossref]

Chembo, Y. K.

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. K. Chembo, and L. Larger, “Slow–fast dynamics of a time-delayed electro-optic oscillator,” Phil. Trans. R. Soc. A 371, 20120459 (2013).
[Crossref]

A. Coillet, R. Henriet, P. Salzenstein, K. P. Huy, L. Larger, and Y. K. Chembo, “Time-domain dynamics and stability analysis of optoelectronic oscillators based on whispering-gallery mode resonators,” IEEE J. Sel. Top. Quantum Electron. 19, 6000112(2013).
[Crossref]

R. M. Nguimdo, Y. K. Chembo, P. Colet, and L. Larger, “On the phase noise performance of nonlinear double-loop optoelectronic microwave oscillators,” IEEE J. Quantum Electron. 48, 1415–1423 (2012).
[Crossref]

R. Martinenghi, S. Rybalko, M. Jacquot, Y. K. Chembo, and L. Larger, “Photonic nonlinear transient computing with multiple-delay wavelength dynamics,” Phys. Rev. Lett. 108, 244101 (2012).
[Crossref]

Y. K. Chembo, A. Hmima, P.-A. Lacourt, L. Larger, and J. M. Dudley, “Generation of ultralow jitter optical pulses using optoelectronic oscillators with time-lens soliton-assisted compression,” J. Lightwave Technol. 27, 5160–5167 (2009).
[Crossref]

Y. K. Chembo, L. Larger, R. Bendoula, and P. Colet, “Effects of gain and bandwidth on the multimode behavior of optoelectronic microwave oscillators,” Opt. Express 16, 9067–9072 (2008).
[Crossref]

Y. K. Chembo, L. Larger, and P. Colet, “Nonlinear dynamics and spectral stability of optoelectronic oscillators,” IEEE J. Quantum Electron. 44, 858–866 (2008).
[Crossref]

Y. K. Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola, and P. Colet, “Dynamic instabilities of microwaves generated with optoelectronic oscillators,” Opt. Lett. 32, 2571–2573 (2007).
[Crossref]

Chen, C.-C.

J.-P. Goedgebuer, P. Levy, L. Larger, C.-C. Chen, and W. T. Rhodes, “Optical communication with synchronized hyperchaos generated electrooptically,” IEEE J. Quantum Electron. 38, 1178–1183 (2002).
[Crossref]

Cohen, A. B.

B. Ravoori, A. B. Cohen, J. Sun, A. E. Motter, T. E. Murphy, and R. Roy, “Robustness of optimal synchronization in real networks,” Phys. Rev. Lett. 107, 034102 (2011).
[Crossref]

A. B. Cohen, B. Ravoori, T. E. Murphy, and R. Roy, “Using synchronization for prediction of high-dimensional chaotic dynamics,” Phys. Rev. Lett. 101, 154102 (2008).
[Crossref]

Coillet, A.

A. Coillet, R. Henriet, P. Salzenstein, K. P. Huy, L. Larger, and Y. K. Chembo, “Time-domain dynamics and stability analysis of optoelectronic oscillators based on whispering-gallery mode resonators,” IEEE J. Sel. Top. Quantum Electron. 19, 6000112(2013).
[Crossref]

Colet, P.

R. M. Nguimdo, Y. K. Chembo, P. Colet, and L. Larger, “On the phase noise performance of nonlinear double-loop optoelectronic microwave oscillators,” IEEE J. Quantum Electron. 48, 1415–1423 (2012).
[Crossref]

Y. K. Chembo, L. Larger, R. Bendoula, and P. Colet, “Effects of gain and bandwidth on the multimode behavior of optoelectronic microwave oscillators,” Opt. Express 16, 9067–9072 (2008).
[Crossref]

Y. K. Chembo, L. Larger, and P. Colet, “Nonlinear dynamics and spectral stability of optoelectronic oscillators,” IEEE J. Quantum Electron. 44, 858–866 (2008).
[Crossref]

Y. K. Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola, and P. Colet, “Dynamic instabilities of microwaves generated with optoelectronic oscillators,” Opt. Lett. 32, 2571–2573 (2007).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438, 343–346 (2005).
[Crossref]

Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, “Chaotic breathers in delayed electro-optical systems,” Phys. Rev. Lett. 95, 203903 (2005).
[Crossref]

D’Huys, O.

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. K. Chembo, and L. Larger, “Slow–fast dynamics of a time-delayed electro-optic oscillator,” Phil. Trans. R. Soc. A 371, 20120459 (2013).
[Crossref]

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Strongly asymmetric square waves in a time-delayed system,” Phys. Rev. E 86, 055201(R) (2012).
[Crossref]

Daido, H.

K. Ikeda, H. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
[Crossref]

Danckaert, J.

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. K. Chembo, and L. Larger, “Slow–fast dynamics of a time-delayed electro-optic oscillator,” Phil. Trans. R. Soc. A 371, 20120459 (2013).
[Crossref]

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Strongly asymmetric square waves in a time-delayed system,” Phys. Rev. E 86, 055201(R) (2012).
[Crossref]

Dudley, J. M.

Erneux, T.

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. K. Chembo, and L. Larger, “Slow–fast dynamics of a time-delayed electro-optic oscillator,” Phil. Trans. R. Soc. A 371, 20120459 (2013).
[Crossref]

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Strongly asymmetric square waves in a time-delayed system,” Phys. Rev. E 86, 055201(R) (2012).
[Crossref]

T. Erneux, Applied Delay Differential Equations (Springer, 2010).

Fischer, I.

M. C. Soriano, J. Garcia-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]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438, 343–346 (2005).
[Crossref]

Gao, Z.

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Scholl, “Broadband chaos generated by an optoelectronic oscillator,” Phys. Rev. Lett. 104, 113901 (2010).
[Crossref]

Garcia-Ojalvo, J.

M. C. Soriano, J. Garcia-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]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438, 343–346 (2005).
[Crossref]

Gastaud, N.

Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, “Chaotic breathers in delayed electro-optical systems,” Phys. Rev. Lett. 95, 203903 (2005).
[Crossref]

Gauthier, D. J.

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Scholl, “Broadband chaos generated by an optoelectronic oscillator,” Phys. Rev. Lett. 104, 113901 (2010).
[Crossref]

Goedgebuer, J.-P.

J.-P. Goedgebuer, P. Levy, L. Larger, C.-C. Chen, and W. T. Rhodes, “Optical communication with synchronized hyperchaos generated electrooptically,” IEEE J. Quantum Electron. 38, 1178–1183 (2002).
[Crossref]

Guckenheimer, J.

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields (Springer-Verlag, 1983).

Henriet, R.

A. Coillet, R. Henriet, P. Salzenstein, K. P. Huy, L. Larger, and Y. K. Chembo, “Time-domain dynamics and stability analysis of optoelectronic oscillators based on whispering-gallery mode resonators,” IEEE J. Sel. Top. Quantum Electron. 19, 6000112(2013).
[Crossref]

Hmima, A.

Holmes, P.

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields (Springer-Verlag, 1983).

Hu, H. Y.

Z. H. Wang and H. Y. Hu, “Pseudo-oscillator analysis of scalar nonlinear time-delay systems near a Hopf bifurcation,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 17, 2805–2814 (2007).
[Crossref]

Huy, K. P.

A. Coillet, R. Henriet, P. Salzenstein, K. P. Huy, L. Larger, and Y. K. Chembo, “Time-domain dynamics and stability analysis of optoelectronic oscillators based on whispering-gallery mode resonators,” IEEE J. Sel. Top. Quantum Electron. 19, 6000112(2013).
[Crossref]

Ikeda, K.

K. Ikeda and M. Matsumoto, “Study of a high-dimensional chaotic attractor,” J. Stat. Phys. 44, 955–983 (1986).
[Crossref]

K. Ikeda, H. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
[Crossref]

K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. 30, 257–261 (1979).
[Crossref]

Illing, L.

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Scholl, “Broadband chaos generated by an optoelectronic oscillator,” Phys. Rev. Lett. 104, 113901 (2010).
[Crossref]

Jacquot, M.

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. K. Chembo, and L. Larger, “Slow–fast dynamics of a time-delayed electro-optic oscillator,” Phil. Trans. R. Soc. A 371, 20120459 (2013).
[Crossref]

R. Martinenghi, S. Rybalko, M. Jacquot, Y. K. Chembo, and L. Larger, “Photonic nonlinear transient computing with multiple-delay wavelength dynamics,” Phys. Rev. Lett. 108, 244101 (2012).
[Crossref]

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Strongly asymmetric square waves in a time-delayed system,” Phys. Rev. E 86, 055201(R) (2012).
[Crossref]

Kouomou, Y. C.

Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, “Chaotic breathers in delayed electro-optical systems,” Phys. Rev. Lett. 95, 203903 (2005).
[Crossref]

Lacourt, P.-A.

Lakshamanan, M.

M. Lakshamanan and D. V. Senthikumar, Dynamics of Nonlinear Time-Delay Systems (Springer, 2011).

Larger, L.

L. Larger, “Complexity in electro-optic delay dynamics: modelling, design and applications,” Phil. Trans. R. Soc. A 371, 20120464 (2013).
[Crossref]

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. K. Chembo, and L. Larger, “Slow–fast dynamics of a time-delayed electro-optic oscillator,” Phil. Trans. R. Soc. A 371, 20120459 (2013).
[Crossref]

A. Coillet, R. Henriet, P. Salzenstein, K. P. Huy, L. Larger, and Y. K. Chembo, “Time-domain dynamics and stability analysis of optoelectronic oscillators based on whispering-gallery mode resonators,” IEEE J. Sel. Top. Quantum Electron. 19, 6000112(2013).
[Crossref]

R. M. Nguimdo, Y. K. Chembo, P. Colet, and L. Larger, “On the phase noise performance of nonlinear double-loop optoelectronic microwave oscillators,” IEEE J. Quantum Electron. 48, 1415–1423 (2012).
[Crossref]

R. Martinenghi, S. Rybalko, M. Jacquot, Y. K. Chembo, and L. Larger, “Photonic nonlinear transient computing with multiple-delay wavelength dynamics,” Phys. Rev. Lett. 108, 244101 (2012).
[Crossref]

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Strongly asymmetric square waves in a time-delayed system,” Phys. Rev. E 86, 055201(R) (2012).
[Crossref]

Y. K. Chembo, A. Hmima, P.-A. Lacourt, L. Larger, and J. M. Dudley, “Generation of ultralow jitter optical pulses using optoelectronic oscillators with time-lens soliton-assisted compression,” J. Lightwave Technol. 27, 5160–5167 (2009).
[Crossref]

Y. K. Chembo, L. Larger, R. Bendoula, and P. Colet, “Effects of gain and bandwidth on the multimode behavior of optoelectronic microwave oscillators,” Opt. Express 16, 9067–9072 (2008).
[Crossref]

Y. K. Chembo, L. Larger, and P. Colet, “Nonlinear dynamics and spectral stability of optoelectronic oscillators,” IEEE J. Quantum Electron. 44, 858–866 (2008).
[Crossref]

Y. K. Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola, and P. Colet, “Dynamic instabilities of microwaves generated with optoelectronic oscillators,” Opt. Lett. 32, 2571–2573 (2007).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438, 343–346 (2005).
[Crossref]

Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, “Chaotic breathers in delayed electro-optical systems,” Phys. Rev. Lett. 95, 203903 (2005).
[Crossref]

J.-P. Goedgebuer, P. Levy, L. Larger, C.-C. Chen, and W. T. Rhodes, “Optical communication with synchronized hyperchaos generated electrooptically,” IEEE J. Quantum Electron. 38, 1178–1183 (2002).
[Crossref]

Levy, P.

J.-P. Goedgebuer, P. Levy, L. Larger, C.-C. Chen, and W. T. Rhodes, “Optical communication with synchronized hyperchaos generated electrooptically,” IEEE J. Quantum Electron. 38, 1178–1183 (2002).
[Crossref]

Maleki, L.

Martinenghi, R.

R. Martinenghi, S. Rybalko, M. Jacquot, Y. K. Chembo, and L. Larger, “Photonic nonlinear transient computing with multiple-delay wavelength dynamics,” Phys. Rev. Lett. 108, 244101 (2012).
[Crossref]

Matsumoto, M.

K. Ikeda and M. Matsumoto, “Study of a high-dimensional chaotic attractor,” J. Stat. Phys. 44, 955–983 (1986).
[Crossref]

Mirasso, C. R.

M. C. Soriano, J. Garcia-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]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438, 343–346 (2005).
[Crossref]

Motter, A. E.

B. Ravoori, A. B. Cohen, J. Sun, A. E. Motter, T. E. Murphy, and R. Roy, “Robustness of optimal synchronization in real networks,” Phys. Rev. Lett. 107, 034102 (2011).
[Crossref]

Murphy, T. E.

B. Ravoori, A. B. Cohen, J. Sun, A. E. Motter, T. E. Murphy, and R. Roy, “Robustness of optimal synchronization in real networks,” Phys. Rev. Lett. 107, 034102 (2011).
[Crossref]

A. B. Cohen, B. Ravoori, T. E. Murphy, and R. Roy, “Using synchronization for prediction of high-dimensional chaotic dynamics,” Phys. Rev. Lett. 101, 154102 (2008).
[Crossref]

Nguimdo, R. M.

R. M. Nguimdo, Y. K. Chembo, P. Colet, and L. Larger, “On the phase noise performance of nonlinear double-loop optoelectronic microwave oscillators,” IEEE J. Quantum Electron. 48, 1415–1423 (2012).
[Crossref]

Pesquera, L.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438, 343–346 (2005).
[Crossref]

Ravoori, B.

B. Ravoori, A. B. Cohen, J. Sun, A. E. Motter, T. E. Murphy, and R. Roy, “Robustness of optimal synchronization in real networks,” Phys. Rev. Lett. 107, 034102 (2011).
[Crossref]

A. B. Cohen, B. Ravoori, T. E. Murphy, and R. Roy, “Using synchronization for prediction of high-dimensional chaotic dynamics,” Phys. Rev. Lett. 101, 154102 (2008).
[Crossref]

Rhodes, W. T.

J.-P. Goedgebuer, P. Levy, L. Larger, C.-C. Chen, and W. T. Rhodes, “Optical communication with synchronized hyperchaos generated electrooptically,” IEEE J. Quantum Electron. 38, 1178–1183 (2002).
[Crossref]

Roy, R.

B. Ravoori, A. B. Cohen, J. Sun, A. E. Motter, T. E. Murphy, and R. Roy, “Robustness of optimal synchronization in real networks,” Phys. Rev. Lett. 107, 034102 (2011).
[Crossref]

A. B. Cohen, B. Ravoori, T. E. Murphy, and R. Roy, “Using synchronization for prediction of high-dimensional chaotic dynamics,” Phys. Rev. Lett. 101, 154102 (2008).
[Crossref]

Rubiola, E.

Rybalko, S.

R. Martinenghi, S. Rybalko, M. Jacquot, Y. K. Chembo, and L. Larger, “Photonic nonlinear transient computing with multiple-delay wavelength dynamics,” Phys. Rev. Lett. 108, 244101 (2012).
[Crossref]

Salzenstein, P.

A. Coillet, R. Henriet, P. Salzenstein, K. P. Huy, L. Larger, and Y. K. Chembo, “Time-domain dynamics and stability analysis of optoelectronic oscillators based on whispering-gallery mode resonators,” IEEE J. Sel. Top. Quantum Electron. 19, 6000112(2013).
[Crossref]

Scholl, E.

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Scholl, “Broadband chaos generated by an optoelectronic oscillator,” Phys. Rev. Lett. 104, 113901 (2010).
[Crossref]

Senthikumar, D. V.

M. Lakshamanan and D. V. Senthikumar, Dynamics of Nonlinear Time-Delay Systems (Springer, 2011).

Shore, K. A.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438, 343–346 (2005).
[Crossref]

Soriano, M. C.

M. C. Soriano, J. Garcia-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]

Sun, J.

B. Ravoori, A. B. Cohen, J. Sun, A. E. Motter, T. E. Murphy, and R. Roy, “Robustness of optimal synchronization in real networks,” Phys. Rev. Lett. 107, 034102 (2011).
[Crossref]

Syvridis, D.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438, 343–346 (2005).
[Crossref]

Tavernier, H.

Wang, Z. H.

Z. H. Wang, “An iteration method for calculating the periodic solution of time-delay systems after a Hopf bifurcation,” Nonlinear Dyn. 53, 1–11 (2008).
[Crossref]

Z. H. Wang and H. Y. Hu, “Pseudo-oscillator analysis of scalar nonlinear time-delay systems near a Hopf bifurcation,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 17, 2805–2814 (2007).
[Crossref]

Weicker, L.

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. K. Chembo, and L. Larger, “Slow–fast dynamics of a time-delayed electro-optic oscillator,” Phil. Trans. R. Soc. A 371, 20120459 (2013).
[Crossref]

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Strongly asymmetric square waves in a time-delayed system,” Phys. Rev. E 86, 055201(R) (2012).
[Crossref]

Yao, X. S.

IEEE J. Quantum Electron. (3)

Y. K. Chembo, L. Larger, and P. Colet, “Nonlinear dynamics and spectral stability of optoelectronic oscillators,” IEEE J. Quantum Electron. 44, 858–866 (2008).
[Crossref]

J.-P. Goedgebuer, P. Levy, L. Larger, C.-C. Chen, and W. T. Rhodes, “Optical communication with synchronized hyperchaos generated electrooptically,” IEEE J. Quantum Electron. 38, 1178–1183 (2002).
[Crossref]

R. M. Nguimdo, Y. K. Chembo, P. Colet, and L. Larger, “On the phase noise performance of nonlinear double-loop optoelectronic microwave oscillators,” IEEE J. Quantum Electron. 48, 1415–1423 (2012).
[Crossref]

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

A. Coillet, R. Henriet, P. Salzenstein, K. P. Huy, L. Larger, and Y. K. Chembo, “Time-domain dynamics and stability analysis of optoelectronic oscillators based on whispering-gallery mode resonators,” IEEE J. Sel. Top. Quantum Electron. 19, 6000112(2013).
[Crossref]

Int. J. Bifurcation Chaos Appl. Sci. Eng. (1)

Z. H. Wang and H. Y. Hu, “Pseudo-oscillator analysis of scalar nonlinear time-delay systems near a Hopf bifurcation,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 17, 2805–2814 (2007).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (1)

J. Stat. Phys. (1)

K. Ikeda and M. Matsumoto, “Study of a high-dimensional chaotic attractor,” J. Stat. Phys. 44, 955–983 (1986).
[Crossref]

Nature (1)

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438, 343–346 (2005).
[Crossref]

Nonlinear Dyn. (1)

Z. H. Wang, “An iteration method for calculating the periodic solution of time-delay systems after a Hopf bifurcation,” Nonlinear Dyn. 53, 1–11 (2008).
[Crossref]

Opt. Commun. (1)

K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. 30, 257–261 (1979).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phil. Trans. R. Soc. A (2)

L. Larger, “Complexity in electro-optic delay dynamics: modelling, design and applications,” Phil. Trans. R. Soc. A 371, 20120464 (2013).
[Crossref]

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. K. Chembo, and L. Larger, “Slow–fast dynamics of a time-delayed electro-optic oscillator,” Phil. Trans. R. Soc. A 371, 20120459 (2013).
[Crossref]

Phys. Rev. E (1)

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Strongly asymmetric square waves in a time-delayed system,” Phys. Rev. E 86, 055201(R) (2012).
[Crossref]

Phys. Rev. Lett. (6)

R. Martinenghi, S. Rybalko, M. Jacquot, Y. K. Chembo, and L. Larger, “Photonic nonlinear transient computing with multiple-delay wavelength dynamics,” Phys. Rev. Lett. 108, 244101 (2012).
[Crossref]

A. B. Cohen, B. Ravoori, T. E. Murphy, and R. Roy, “Using synchronization for prediction of high-dimensional chaotic dynamics,” Phys. Rev. Lett. 101, 154102 (2008).
[Crossref]

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Scholl, “Broadband chaos generated by an optoelectronic oscillator,” Phys. Rev. Lett. 104, 113901 (2010).
[Crossref]

B. Ravoori, A. B. Cohen, J. Sun, A. E. Motter, T. E. Murphy, and R. Roy, “Robustness of optimal synchronization in real networks,” Phys. Rev. Lett. 107, 034102 (2011).
[Crossref]

Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, “Chaotic breathers in delayed electro-optical systems,” Phys. Rev. Lett. 95, 203903 (2005).
[Crossref]

K. Ikeda, H. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
[Crossref]

Rev. Mod. Phys. (1)

M. C. Soriano, J. Garcia-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]

Other (3)

T. Erneux, Applied Delay Differential Equations (Springer, 2010).

M. Lakshamanan and D. V. Senthikumar, Dynamics of Nonlinear Time-Delay Systems (Springer, 2011).

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields (Springer-Verlag, 1983).

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Figures (9)

Fig. 1.
Fig. 1.

Experimental setup: LD, laser diode; MZM, Mach–Zehnder modulator; PD, photodiode; PC, polarization controller; BPF, bandpass filter.

Fig. 2.
Fig. 2.

Bifurcation diagram of Eq. (2) for ϕ=π/4 and TD=0. The solid line is obtained analytically from Eq. (14), while the dotted line is obtained from the numerical simulation of Eq. (2). The labels (1) and (2) indicate the two dynamical regimes.

Fig. 3.
Fig. 3.

Quasi-sinusoidal temporal evolution of the OEO after the Hopf bifurcation depicted in Fig. 2 (T=10μs). (a) Experimental result (pump power set just above the threshold at Pth1.2mW). (b) Numerical simulation of Eq. (2) for β=1.032. Note that close to the bifurcation, the experimental and numerical plots [parts (a) and (b), respectively], display a period comparable to the theoretical value provided by Eq. (4), that is T0=2π/Ω0=2πθTD250μs.

Fig. 4.
Fig. 4.

Typical experimental timetraces of the relaxation-like slow-scale dynamics of the OEO far away from the Hopf bifurcation. The period Texp is found to vary significantly with the gain, still having the same order of magnitude as the theoretical Hopf period T0. (Left) Texp560μs. (Right) Texp760μs.

Fig. 5.
Fig. 5.

Typical time evolution of x in Eq. (15) for β>1, ϕ=π/4, and TD=0. Note the symmetric nature of this numerical timetrace.

Fig. 6.
Fig. 6.

Evolution of the period of the symmetric wave oscillation for TD=0 and ϕ=π/4. The solid line is obtained with the analytical simulation of Eq. (23), while the dots stand for the numerical simulations of Eq. (15).

Fig. 7.
Fig. 7.

Temporal evolution of Eq. (15) for ϕ=0.51 and TD=0. Note the asymmetric nature of this numerical timetrace.

Fig. 8.
Fig. 8.

Plot of the xz phase space, for ϕ=0.51 and TD=0. The blue continuous line is the numerical simulation of Eq. (15), and the red-dotted line is the numerical simulation of Eq. (16).

Fig. 9.
Fig. 9.

Evolution of the period of the asymmetric wave oscillation and TD=0. The solid line is obtained by the analytical simulation of Eq. (28), while the dots stand for the numerical simulation of Eq. (15).

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

dxdt=F[t,x(t),x(tTD)],
x+τdxdt+1θt0tx(s)ds=βcos2[x(tTD)+ϕ],
γ0=112TDθ,
Ω0=1θTD.
x(t)=Acos[Ω0t+ψ]=12AeiΩ0t+c.c.,
eizcosα=n=+inJn(z)einα,
cos2[x(tTD)+ϕ]=C0+n=1+{12CneinΩ0t+c.c.},
C0=12+12J0(2A)cos2ϕ,Cn=12Jn(2A)inein(ψσ){e2iϕ+(1)ne2iϕ},n0.
[1+iΩ0τ+1iΩ0θ]A=βC1,
[1iσ]A=βsin2ϕJ1(2A)eiψeiσ,
eiσ1iσ1σ22=112TDθ,
Jc1(2A)=12γ[112TDθ],
12=12γ0[112TDθ],
A=3[1134|γ|[112TDθ]1]1/2,
y˙=x,τx˙=x1θy+β{cos2[x+ϕ]cos2(ϕ)}.
z=x+β{cos2[x+ϕ]cos2(ϕ)},
x˙=1θx1+βsin(2x+2ϕ).
1+βsin(2x+2ϕ)=0.
xN=xQ=12arccos(1|γ|).
xM=xP=3[1134|γ|1]1/2.
TSym=TMN+TPQ,
TMN=θxMxN1+γcos(2x)xdx.
TSym(γ)=2θ[ln(xMxN)+γ[Ci(2xM)Ci(2xN)]],
TAsym=TAB+TCD
TAB=xAxBf(x)dx,TCD=xCxDf(x)dx,
f(x)=θ1+βsin(2x+2ϕ)x.
TAsym(ϕ)=θln(xBxDxAxC)θβsin2ϕ[Ci(2xB)+Ci(2xD)Ci(2xA)Ci(2xC)]+θβcos2ϕ[Si(2xB)+Si(2xD)Si(2xA)Si(2xC)],
xB=12arcsin(1β)ϕ,xD=12[π+arcsin(1β)]ϕ.
xA+β[cos2(xA+ϕ)cos2(ϕ)]=z(xD),xC+β[cos2(xC+ϕ)cos2(ϕ)]=z(xB).
dx(t1)dt1βcos(2x(t1ϵ0T))dx(t1ϵ0TD)dt1+x(t1)θϵ0=0,
x(t1ϵ0TD)x(t1)+η(ϵ0,TD,xmax),
d2x(t2)dt22+dx(t2)dt2βcos[2x(t2TD/ϵ0)]×dx(t2TD/ε0)dt2=0,

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