Abstract

In the dynamics of optical systems, one commonly needs to cope with the problem of coexisting deterministic and stochastic components. The separation of these components is an important, although difficult, task. Often the time scales at which determinism and noise dominate the system’s dynamics differ. In this Letter we propose to use information-theory-derived quantifiers, more precisely, permutation entropy and statistical complexity, to distinguish between the two behaviors. Based on experiments of a paradigmatic opto-electronic oscillator, we demonstrate that the time scales at which deterministic or noisy behavior dominate can be identified. Supporting numerical simulations prove the accuracy of this identification.

© 2011 Optical Society of America

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  1. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980).
    [Crossref]
  2. T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, Phys. Rev. Lett. 86, 795 (2001).
    [Crossref] [PubMed]
  3. L. Larger and J. M. Dudley, Nature 465, 41 (2010).
    [Crossref] [PubMed]
  4. K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, Phys. Rev. Lett. 104, 113901 (2010).
    [Crossref] [PubMed]
  5. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 438, 343 (2005).
    [Crossref] [PubMed]
  6. Y. K. Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola, and P. Colet, Opt. Lett. 32, 2571 (2007).
    [Crossref]
  7. M. Peil, M. Jacquot, Y. C. Kouomou, L. Larger, and T. Erneux, Phys. Rev. E 79, 026208 (2009).
    [Crossref]
  8. O. A. Rosso, H. A. Larrondo, M. T. Martin, A. Plastino, and M. A. Fuentes, Phys. Rev. Lett. 99, 154102 (2007).
    [Crossref] [PubMed]
  9. M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, IEEE J. Quantum Electron. 47, 252 (2011).
    [Crossref]
  10. L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, Phys. Rev. E 82, 046212 (2010).
    [Crossref]
  11. C. Bandt and B. Pompe, Phys. Rev. Lett. 88, 174102 (2002).
    [Crossref] [PubMed]
  12. K. Ikeda, Opt. Commun. 30, 257 (1979).
    [Crossref]

2011 (1)

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, IEEE J. Quantum Electron. 47, 252 (2011).
[Crossref]

2010 (3)

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, Phys. Rev. E 82, 046212 (2010).
[Crossref]

L. Larger and J. M. Dudley, Nature 465, 41 (2010).
[Crossref] [PubMed]

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, Phys. Rev. Lett. 104, 113901 (2010).
[Crossref] [PubMed]

2009 (1)

M. Peil, M. Jacquot, Y. C. Kouomou, L. Larger, and T. Erneux, Phys. Rev. E 79, 026208 (2009).
[Crossref]

2007 (2)

O. A. Rosso, H. A. Larrondo, M. T. Martin, A. Plastino, and M. A. Fuentes, Phys. Rev. Lett. 99, 154102 (2007).
[Crossref] [PubMed]

Y. K. Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola, and P. Colet, Opt. Lett. 32, 2571 (2007).
[Crossref]

2005 (1)

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 438, 343 (2005).
[Crossref] [PubMed]

2002 (1)

C. Bandt and B. Pompe, Phys. Rev. Lett. 88, 174102 (2002).
[Crossref] [PubMed]

2001 (1)

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, Phys. Rev. Lett. 86, 795 (2001).
[Crossref] [PubMed]

1980 (1)

R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980).
[Crossref]

1979 (1)

K. Ikeda, Opt. Commun. 30, 257 (1979).
[Crossref]

Annovazzi-Lodi, V.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 438, 343 (2005).
[Crossref] [PubMed]

Argyris, A.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 438, 343 (2005).
[Crossref] [PubMed]

Bandt, C.

C. Bandt and B. Pompe, Phys. Rev. Lett. 88, 174102 (2002).
[Crossref] [PubMed]

Bendoula, R.

Callan, K. E.

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, Phys. Rev. Lett. 104, 113901 (2010).
[Crossref] [PubMed]

Chembo, Y. K.

Colet, P.

Y. K. Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola, and P. Colet, Opt. Lett. 32, 2571 (2007).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 438, 343 (2005).
[Crossref] [PubMed]

Dudley, J. M.

L. Larger and J. M. Dudley, Nature 465, 41 (2010).
[Crossref] [PubMed]

Elsässer, W.

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, Phys. Rev. Lett. 86, 795 (2001).
[Crossref] [PubMed]

Erneux, T.

M. Peil, M. Jacquot, Y. C. Kouomou, L. Larger, and T. Erneux, Phys. Rev. E 79, 026208 (2009).
[Crossref]

Fischer, I.

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, IEEE J. Quantum Electron. 47, 252 (2011).
[Crossref]

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, Phys. Rev. E 82, 046212 (2010).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 438, 343 (2005).
[Crossref] [PubMed]

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, Phys. Rev. Lett. 86, 795 (2001).
[Crossref] [PubMed]

Fuentes, M. A.

O. A. Rosso, H. A. Larrondo, M. T. Martin, A. Plastino, and M. A. Fuentes, Phys. Rev. Lett. 99, 154102 (2007).
[Crossref] [PubMed]

Gao, Z.

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, Phys. Rev. Lett. 104, 113901 (2010).
[Crossref] [PubMed]

García-Ojalvo, J.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 438, 343 (2005).
[Crossref] [PubMed]

Gauthier, D. J.

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, Phys. Rev. Lett. 104, 113901 (2010).
[Crossref] [PubMed]

Heil, T.

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, Phys. Rev. Lett. 86, 795 (2001).
[Crossref] [PubMed]

Ikeda, K.

K. Ikeda, Opt. Commun. 30, 257 (1979).
[Crossref]

Illing, L.

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, Phys. Rev. Lett. 104, 113901 (2010).
[Crossref] [PubMed]

Jacquot, M.

M. Peil, M. Jacquot, Y. C. Kouomou, L. Larger, and T. Erneux, Phys. Rev. E 79, 026208 (2009).
[Crossref]

Kobayashi, K.

R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980).
[Crossref]

Kouomou, Y. C.

M. Peil, M. Jacquot, Y. C. Kouomou, L. Larger, and T. Erneux, Phys. Rev. E 79, 026208 (2009).
[Crossref]

Lang, R.

R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980).
[Crossref]

Larger, L.

L. Larger and J. M. Dudley, Nature 465, 41 (2010).
[Crossref] [PubMed]

M. Peil, M. Jacquot, Y. C. Kouomou, L. Larger, and T. Erneux, Phys. Rev. E 79, 026208 (2009).
[Crossref]

Y. K. Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola, and P. Colet, Opt. Lett. 32, 2571 (2007).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 438, 343 (2005).
[Crossref] [PubMed]

Larrondo, H. A.

O. A. Rosso, H. A. Larrondo, M. T. Martin, A. Plastino, and M. A. Fuentes, Phys. Rev. Lett. 99, 154102 (2007).
[Crossref] [PubMed]

Martin, M. T.

O. A. Rosso, H. A. Larrondo, M. T. Martin, A. Plastino, and M. A. Fuentes, Phys. Rev. Lett. 99, 154102 (2007).
[Crossref] [PubMed]

Mirasso, C. R.

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, IEEE J. Quantum Electron. 47, 252 (2011).
[Crossref]

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, Phys. Rev. E 82, 046212 (2010).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 438, 343 (2005).
[Crossref] [PubMed]

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, Phys. Rev. Lett. 86, 795 (2001).
[Crossref] [PubMed]

Mulet, J.

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, Phys. Rev. Lett. 86, 795 (2001).
[Crossref] [PubMed]

Peil, M.

M. Peil, M. Jacquot, Y. C. Kouomou, L. Larger, and T. Erneux, Phys. Rev. E 79, 026208 (2009).
[Crossref]

Pesquera, L.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 438, 343 (2005).
[Crossref] [PubMed]

Plastino, A.

O. A. Rosso, H. A. Larrondo, M. T. Martin, A. Plastino, and M. A. Fuentes, Phys. Rev. Lett. 99, 154102 (2007).
[Crossref] [PubMed]

Pompe, B.

C. Bandt and B. Pompe, Phys. Rev. Lett. 88, 174102 (2002).
[Crossref] [PubMed]

Rosso, O. A.

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, IEEE J. Quantum Electron. 47, 252 (2011).
[Crossref]

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, Phys. Rev. E 82, 046212 (2010).
[Crossref]

O. A. Rosso, H. A. Larrondo, M. T. Martin, A. Plastino, and M. A. Fuentes, Phys. Rev. Lett. 99, 154102 (2007).
[Crossref] [PubMed]

Rubiola, E.

Schöll, E.

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, Phys. Rev. Lett. 104, 113901 (2010).
[Crossref] [PubMed]

Shore, K. A.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 438, 343 (2005).
[Crossref] [PubMed]

Soriano, M. C.

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, IEEE J. Quantum Electron. 47, 252 (2011).
[Crossref]

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, Phys. Rev. E 82, 046212 (2010).
[Crossref]

Syvridis, D.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 438, 343 (2005).
[Crossref] [PubMed]

Tavernier, H.

Zunino, L.

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, IEEE J. Quantum Electron. 47, 252 (2011).
[Crossref]

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, Phys. Rev. E 82, 046212 (2010).
[Crossref]

IEEE J. Quantum Electron. (2)

R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980).
[Crossref]

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, IEEE J. Quantum Electron. 47, 252 (2011).
[Crossref]

Nature (2)

L. Larger and J. M. Dudley, Nature 465, 41 (2010).
[Crossref] [PubMed]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 438, 343 (2005).
[Crossref] [PubMed]

Opt. Commun. (1)

K. Ikeda, Opt. Commun. 30, 257 (1979).
[Crossref]

Opt. Lett. (1)

Phys. Rev. E (2)

M. Peil, M. Jacquot, Y. C. Kouomou, L. Larger, and T. Erneux, Phys. Rev. E 79, 026208 (2009).
[Crossref]

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, Phys. Rev. E 82, 046212 (2010).
[Crossref]

Phys. Rev. Lett. (4)

C. Bandt and B. Pompe, Phys. Rev. Lett. 88, 174102 (2002).
[Crossref] [PubMed]

O. A. Rosso, H. A. Larrondo, M. T. Martin, A. Plastino, and M. A. Fuentes, Phys. Rev. Lett. 99, 154102 (2007).
[Crossref] [PubMed]

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, Phys. Rev. Lett. 104, 113901 (2010).
[Crossref] [PubMed]

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, Phys. Rev. Lett. 86, 795 (2001).
[Crossref] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

The opto-electronic oscillator is composed of a semiconductor laser diode feeding a Mach-Zehnder (MZ) modulator that performs a sine squared nonlinear transformation, an optical delay line, and an opto-electronic feedback for intensity detection, linear filtering, and amplification. This feedback serves as the drive of the MZ modulator, closing the delay loop (delay time 21 μs ).

Fig. 2
Fig. 2

Power spectrum of the experimentally recorded time-traces of the delayed opto-electronic oscillator operating in the chaotic regime.

Fig. 3
Fig. 3

Permutation entropy (H, circles) and permutation statistical complexity (C, crosses) for an experimental time series recorded in the chaotic regime as functions of τ. The inset represents the pair entropy complexity at each embedding delay [8]. Embedding dimension D = 6 and sampling rate 500   MSamples / s are used.

Fig. 4
Fig. 4

Values of the permutation entropy and statistical complexity as functions of the embedding delay in the H C plane for (a) original experimental time series [circles] and corresponding moving averages of different lengths [squares correspond to 20 ns , stars to 40 ns , and crosses to 160 ns length], and (b) numerical time series without noise [crosses], with added noise [circles], and corresponding moving averages of different lengths [squares correspond to 20 ns and stars to 40 ns length]. Parameters, β = 3.6 , T = 87.2 , Φ = 0.85 rad .

Equations (1)

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d x ( t ) d t = x ( t ) + β sin 2 [ x ( t T ) + Φ ] ,

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